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Theorem ioorval 25545

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 2738	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
2		infeq1 9498	. . . 4 ⊢ (𝑥 = 𝐴 → inf(𝑥, ℝ^, < ) = inf(𝐴, ℝ^, < ))
3		supeq1 9467	. . . 4 ⊢ (𝑥 = 𝐴 → sup(𝑥, ℝ^, < ) = sup(𝐴, ℝ^, < ))
4	2, 3	opeq12d 4861	. . 3 ⊢ (𝑥 = 𝐴 → ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩ = ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩)
5	1, 4	ifbieq2d 4532	. 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 5449	. . 3 ⊢ ⟨0, 0⟩ ∈ V
8		opex 5449	. . 3 ⊢ ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩ ∈ V
9	7, 8	ifex 4556	. 2 ⊢ if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩) ∈ V
10	5, 6, 9	fvmpt 6996	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 1539 ∈ wcel 2107 ∅c0 4313 ifcif 4505 ⟨cop 4612 ↦ cmpt 5205 ran crn 5666 ‘cfv 6541 supcsup 9462 infcinf 9463 0cc0 11137 ℝ^*cxr 11276 < clt 11277 (,)cioo 13369

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pr 5412

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ral 3051 df-rex 3060 df-rab 3420 df-v 3465 df-dif 3934 df-un 3936 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-iota 6494 df-fun 6543 df-fv 6549 df-sup 9464 df-inf 9465

This theorem is referenced by: ioorinv2 25546 ioorinv 25547 ioorcl 25548

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