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Theorem suppimacnvss 8172

Description: The support of functions "defined" by inverse images is a subset of the support defined by df-supp 8161. (Contributed by AV, 7-Apr-2019.)

**Assertion**
Ref	Expression
suppimacnvss	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (^◡𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))

Proof of Theorem suppimacnvss Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		exsimpl 1864	. . . . 5 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍) → ∃𝑦 𝑥𝑅𝑦)
2		pm5.1 823	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍) → (𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))
3	2	eximi 1830	. . . . 5 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍) → ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))
4	1, 3	jca 511	. . . 4 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍)))
5	4	a1i 11	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))))
6	5	ss2abdv 4057	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)} ⊆ {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))})
7		cnvimadfsn 8171	. . 3 ⊢ (^◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)}
8	7	a1i 11	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (^◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)})
9		suppvalbr 8164	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))})
10	6, 8, 9	3sstr4d 4026	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (^◡𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 {cab 2705 ≠ wne 2936 Vcvv 3470 ∖ cdif 3942 ⊆ wss 3945 {csn 4625 class class class wbr 5143 ^◡ccnv 5672 “ cima 5676 (class class class)co 7415 supp csupp 8160

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-sep 5294 ax-nul 5301 ax-pr 5424 ax-un 7735

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 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-supp 8161

This theorem is referenced by: suppimacnv 8173 fsuppinisegfi 32462

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