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 25714

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 2767	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
2		infeq1 9438	. . . 4 ⊢ (𝑥 = 𝐴 → inf(𝑥, ℝ^, < ) = inf(𝐴, ℝ^, < ))
3		supeq1 9406	. . . 4 ⊢ (𝑥 = 𝐴 → sup(𝑥, ℝ^, < ) = sup(𝐴, ℝ^, < ))
4	2, 3	opeq12d 4847	. . 3 ⊢ (𝑥 = 𝐴 → ⟨inf(𝑥, ℝ^, < ), sup(𝑥, ℝ^, < )⟩ = ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩)
5	1, 4	ifbieq2d 4515	. 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 5447	. . 3 ⊢ ⟨0, 0⟩ ∈ V
8		opex 5447	. . 3 ⊢ ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩ ∈ V
9	7, 8	ifex 4539	. 2 ⊢ if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ^, < ), sup(𝐴, ℝ^, < )⟩) ∈ V
10	5, 6, 9	fvmpt 6991	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 1570 ∈ wcel 2143 ∅c0 4287 ifcif 4488 ⟨cop 4596 ↦ cmpt 5193 ran crn 5664 ‘cfv 6538 supcsup 9401 infcinf 9402 0cc0 11101 ℝ^*cxr 11243 < clt 11244 (,)cioo 13373

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-pr 5406

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-iota 6494 df-fun 6540 df-fv 6546 df-sup 9403 df-inf 9404

This theorem is referenced by: ioorinv2 25715 ioorinv 25716 ioorcl 25717

W3C validator