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Theorem ioorval 25491

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 2733	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
2		infeq1 9386	. . . 4 ⊢ (𝑥 = 𝐴 → inf(𝑥, ℝ^, < ) = inf(𝐴, ℝ^, < ))
3		supeq1 9354	. . . 4 ⊢ (𝑥 = 𝐴 → sup(𝑥, ℝ^, < ) = sup(𝐴, ℝ^, < ))
4	2, 3	opeq12d 4835	. . 3 ⊢ (𝑥 = 𝐴 → ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩ = ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩)
5	1, 4	ifbieq2d 4505	. 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 5411	. . 3 ⊢ ⟨0, 0⟩ ∈ V
8		opex 5411	. . 3 ⊢ ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩ ∈ V
9	7, 8	ifex 4529	. 2 ⊢ if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩) ∈ V
10	5, 6, 9	fvmpt 6934	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 1540 ∈ wcel 2109 ∅c0 4286 ifcif 4478 ⟨cop 4585 ↦ cmpt 5176 ran crn 5624 ‘cfv 6486 supcsup 9349 infcinf 9350 0cc0 11028 ℝ^*cxr 11167 < clt 11168 (,)cioo 13266

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-iota 6442 df-fun 6488 df-fv 6494 df-sup 9351 df-inf 9352

This theorem is referenced by: ioorinv2 25492 ioorinv 25493 ioorcl 25494

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