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Theorem ioorval 25091
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 = (π‘₯ ∈ ran (,) ↦ if(π‘₯ = βˆ…, ⟨0, 0⟩, ⟨inf(π‘₯, ℝ*, < ), sup(π‘₯, ℝ*, < )⟩))
Assertion
Ref Expression
ioorval (𝐴 ∈ ran (,) β†’ (πΉβ€˜π΄) = if(𝐴 = βˆ…, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩))
Distinct variable group:   π‘₯,𝐴
Allowed substitution hint:   𝐹(π‘₯)

Proof of Theorem ioorval
StepHypRef Expression
1 eqeq1 2737 . . 3 (π‘₯ = 𝐴 β†’ (π‘₯ = βˆ… ↔ 𝐴 = βˆ…))
2 infeq1 9471 . . . 4 (π‘₯ = 𝐴 β†’ inf(π‘₯, ℝ*, < ) = inf(𝐴, ℝ*, < ))
3 supeq1 9440 . . . 4 (π‘₯ = 𝐴 β†’ sup(π‘₯, ℝ*, < ) = sup(𝐴, ℝ*, < ))
42, 3opeq12d 4882 . . 3 (π‘₯ = 𝐴 β†’ ⟨inf(π‘₯, ℝ*, < ), sup(π‘₯, ℝ*, < )⟩ = ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩)
51, 4ifbieq2d 4555 . 2 (π‘₯ = 𝐴 β†’ if(π‘₯ = βˆ…, ⟨0, 0⟩, ⟨inf(π‘₯, ℝ*, < ), sup(π‘₯, ℝ*, < )⟩) = if(𝐴 = βˆ…, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩))
6 ioorf.1 . 2 𝐹 = (π‘₯ ∈ ran (,) ↦ if(π‘₯ = βˆ…, ⟨0, 0⟩, ⟨inf(π‘₯, ℝ*, < ), sup(π‘₯, ℝ*, < )⟩))
7 opex 5465 . . 3 ⟨0, 0⟩ ∈ V
8 opex 5465 . . 3 ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩ ∈ V
97, 8ifex 4579 . 2 if(𝐴 = βˆ…, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩) ∈ V
105, 6, 9fvmpt 6999 1 (𝐴 ∈ ran (,) β†’ (πΉβ€˜π΄) = if(𝐴 = βˆ…, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ…c0 4323  ifcif 4529  βŸ¨cop 4635   ↦ cmpt 5232  ran crn 5678  β€˜cfv 6544  supcsup 9435  infcinf 9436  0cc0 11110  β„*cxr 11247   < clt 11248  (,)cioo 13324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-sup 9437  df-inf 9438
This theorem is referenced by:  ioorinv2  25092  ioorinv  25093  ioorcl  25094
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