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Theorem ioorval 23562
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 (,) → (𝐹𝐴) = if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem ioorval
StepHypRef Expression
1 eqeq1 2764 . . 3 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
2 infeq1 8549 . . . 4 (𝑥 = 𝐴 → inf(𝑥, ℝ*, < ) = inf(𝐴, ℝ*, < ))
3 supeq1 8518 . . . 4 (𝑥 = 𝐴 → sup(𝑥, ℝ*, < ) = sup(𝐴, ℝ*, < ))
42, 3opeq12d 4561 . . 3 (𝑥 = 𝐴 → ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩ = ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩)
51, 4ifbieq2d 4255 . 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 5081 . . 3 ⟨0, 0⟩ ∈ V
8 opex 5081 . . 3 ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩ ∈ V
97, 8ifex 4300 . 2 if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩) ∈ V
105, 6, 9fvmpt 6445 1 (𝐴 ∈ ran (,) → (𝐹𝐴) = if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  c0 4058  ifcif 4230  cop 4327  cmpt 4881  ran crn 5267  cfv 6049  supcsup 8513  infcinf 8514  0cc0 10148  *cxr 10285   < clt 10286  (,)cioo 12388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-sup 8515  df-inf 8516
This theorem is referenced by:  ioorinv2  23563  ioorinv  23564  ioorcl  23565
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