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Theorem ioorval 25543

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 2741	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
2		infeq1 9392	. . . 4 ⊢ (𝑥 = 𝐴 → inf(𝑥, ℝ^, < ) = inf(𝐴, ℝ^, < ))
3		supeq1 9360	. . . 4 ⊢ (𝑥 = 𝐴 → sup(𝑥, ℝ^, < ) = sup(𝐴, ℝ^, < ))
4	2, 3	opeq12d 4839	. . 3 ⊢ (𝑥 = 𝐴 → ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩ = ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩)
5	1, 4	ifbieq2d 4508	. 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 5419	. . 3 ⊢ ⟨0, 0⟩ ∈ V
8		opex 5419	. . 3 ⊢ ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩ ∈ V
9	7, 8	ifex 4532	. 2 ⊢ if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩) ∈ V
10	5, 6, 9	fvmpt 6949	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 1542 ∈ wcel 2114 ∅c0 4287 ifcif 4481 ⟨cop 4588 ↦ cmpt 5181 ran crn 5633 ‘cfv 6500 supcsup 9355 infcinf 9356 0cc0 11038 ℝ^*cxr 11177 < clt 11178 (,)cioo 13273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-pr 5379

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-iota 6456 df-fun 6502 df-fv 6508 df-sup 9357 df-inf 9358

This theorem is referenced by: ioorinv2 25544 ioorinv 25545 ioorcl 25546

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