ioorval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ioorval		Structured version Visualization version GIF version

Theorem ioorval 25490

Description: Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.)

**Hypothesis**
Ref	Expression
ioorf.1	⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩))

**Assertion**
Ref	Expression
ioorval	⊢ (𝐴 ∈ ran (,) → (𝐹‘𝐴) = if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩))

Distinct variable group: 𝑥,𝐴 Allowed substitution hint: 𝐹(𝑥)

**Proof of Theorem ioorval**
Step	Hyp	Ref	Expression
1		eqeq1 2731	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
2		infeq1 9491	. . . 4 ⊢ (𝑥 = 𝐴 → inf(𝑥, ℝ^, < ) = inf(𝐴, ℝ^, < ))
3		supeq1 9460	. . . 4 ⊢ (𝑥 = 𝐴 → sup(𝑥, ℝ^, < ) = sup(𝐴, ℝ^, < ))
4	2, 3	opeq12d 4877	. . 3 ⊢ (𝑥 = 𝐴 → ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩ = ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩)
5	1, 4	ifbieq2d 4550	. 2 ⊢ (𝑥 = 𝐴 → if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩) = if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩))
6		ioorf.1	. 2 ⊢ 𝐹 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, ⟨0, 0⟩, ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩))
7		opex 5460	. . 3 ⊢ ⟨0, 0⟩ ∈ V
8		opex 5460	. . 3 ⊢ ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩ ∈ V
9	7, 8	ifex 4574	. 2 ⊢ if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩) ∈ V
10	5, 6, 9	fvmpt 6999	1 ⊢ (𝐴 ∈ ran (,) → (𝐹‘𝐴) = if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∅c0 4318 ifcif 4524 ⟨cop 4630 ↦ cmpt 5225 ran crn 5673 ‘cfv 6542 supcsup 9455 infcinf 9456 0cc0 11130 ℝ^*cxr 11269 < clt 11270 (,)cioo 13348

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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423

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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-sup 9457 df-inf 9458

This theorem is referenced by: ioorinv2 25491 ioorinv 25492 ioorcl 25493

W3C validator