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Theorem ioorval 24274
 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 2762 . . 3 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
2 infeq1 8973 . . . 4 (𝑥 = 𝐴 → inf(𝑥, ℝ*, < ) = inf(𝐴, ℝ*, < ))
3 supeq1 8942 . . . 4 (𝑥 = 𝐴 → sup(𝑥, ℝ*, < ) = sup(𝐴, ℝ*, < ))
42, 3opeq12d 4771 . . 3 (𝑥 = 𝐴 → ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩ = ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩)
51, 4ifbieq2d 4446 . 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 5324 . . 3 ⟨0, 0⟩ ∈ V
8 opex 5324 . . 3 ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩ ∈ V
97, 8ifex 4470 . 2 if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩) ∈ V
105, 6, 9fvmpt 6759 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 1538   ∈ wcel 2111  ∅c0 4225  ifcif 4420  ⟨cop 4528   ↦ cmpt 5112  ran crn 5525  ‘cfv 6335  supcsup 8937  infcinf 8938  0cc0 10575  ℝ*cxr 10712   < clt 10713  (,)cioo 12779 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-sup 8939  df-inf 8940 This theorem is referenced by:  ioorinv2  24275  ioorinv  24276  ioorcl  24277
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