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Theorem ioorval 25090
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 2736 . . 3 (π‘₯ = 𝐴 β†’ (π‘₯ = βˆ… ↔ 𝐴 = βˆ…))
2 infeq1 9470 . . . 4 (π‘₯ = 𝐴 β†’ inf(π‘₯, ℝ*, < ) = inf(𝐴, ℝ*, < ))
3 supeq1 9439 . . . 4 (π‘₯ = 𝐴 β†’ sup(π‘₯, ℝ*, < ) = sup(𝐴, ℝ*, < ))
42, 3opeq12d 4881 . . 3 (π‘₯ = 𝐴 β†’ ⟨inf(π‘₯, ℝ*, < ), sup(π‘₯, ℝ*, < )⟩ = ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩)
51, 4ifbieq2d 4554 . 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 5464 . . 3 ⟨0, 0⟩ ∈ V
8 opex 5464 . . 3 ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩ ∈ V
97, 8ifex 4578 . 2 if(𝐴 = βˆ…, ⟨0, 0⟩, ⟨inf(𝐴, ℝ*, < ), sup(𝐴, ℝ*, < )⟩) ∈ V
105, 6, 9fvmpt 6998 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 1541   ∈ wcel 2106  βˆ…c0 4322  ifcif 4528  βŸ¨cop 4634   ↦ cmpt 5231  ran crn 5677  β€˜cfv 6543  supcsup 9434  infcinf 9435  0cc0 11109  β„*cxr 11246   < clt 11247  (,)cioo 13323
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-sup 9436  df-inf 9437
This theorem is referenced by:  ioorinv2  25091  ioorinv  25092  ioorcl  25093
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