Theorem fvresex 7963

Description: Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypothesis

Ref	Expression
fvresex.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
fvresex	⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)} ∈ V

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem fvresex

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssv 4004	. . . . . . . 8 ⊢ 𝐴 ⊆ V
2		resmpt 6041	. . . . . . . 8 ⊢ (𝐴 ⊆ V → ((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴) = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ ((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴) = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))
4	3	fveq1i 6898	. . . . . 6 ⊢ (((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴)‘𝑥) = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)
5		fveq2 6897	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
6		eqid 2728	. . . . . . . . 9 ⊢ (𝑧 ∈ V ↦ (𝐹‘𝑧)) = (𝑧 ∈ V ↦ (𝐹‘𝑧))
7		fvex 6910	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
8	5, 6, 7	fvmpt 7005	. . . . . . . 8 ⊢ (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥))
9	8	elv 3477	. . . . . . 7 ⊢ ((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥)
10		fveqres 6944	. . . . . . 7 ⊢ (((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥) → (((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑥))
11	9, 10	ax-mp 5	. . . . . 6 ⊢ (((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑥)
12	4, 11	eqtr3i 2758	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑥)
13	12	eqeq2i 2741	. . . 4 ⊢ (𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) ↔ 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥))
14	13	exbii 1843	. . 3 ⊢ (∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) ↔ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥))
15	14	abbii 2798	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)} = {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)}
16		fvresex.1	. . . 4 ⊢ 𝐴 ∈ V
17	16	mptex 7235	. . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) ∈ V
18	17	fvclex 7962	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)} ∈ V
19	15, 18	eqeltrri 2826	1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)} ∈ V

Syntax hints:   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2705  Vcvv 3471   ⊆ wss 3947   ↦ cmpt 5231   ↾ cres 5680  ‘cfv 6548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556

This theorem is referenced by: (None)