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Theorem ioorval 24102
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 2822 . . 3 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
2 infeq1 8928 . . . 4 (𝑥 = 𝐴 → inf(𝑥, ℝ*, < ) = inf(𝐴, ℝ*, < ))
3 supeq1 8897 . . . 4 (𝑥 = 𝐴 → sup(𝑥, ℝ*, < ) = sup(𝐴, ℝ*, < ))
42, 3opeq12d 4803 . . 3 (𝑥 = 𝐴 → ⟨inf(𝑥, ℝ*, < ), sup(𝑥, ℝ*, < )⟩ = ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩)
51, 4ifbieq2d 4488 . 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 5347 . . 3 ⟨0, 0⟩ ∈ V
8 opex 5347 . . 3 ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩ ∈ V
97, 8ifex 4511 . 2 if(𝐴 = ∅, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩) ∈ V
105, 6, 9fvmpt 6761 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 1528  wcel 2105  c0 4288  ifcif 4463  cop 4563  cmpt 5137  ran crn 5549  cfv 6348  supcsup 8892  infcinf 8893  0cc0 10525  *cxr 10662   < clt 10663  (,)cioo 12726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-sup 8894  df-inf 8895
This theorem is referenced by:  ioorinv2  24103  ioorinv  24104  ioorcl  24105
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