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Theorem opncldf2 23077
Description: The values of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
opncldf.1 𝑋 = 𝐽
opncldf.2 𝐹 = (𝑢𝐽 ↦ (𝑋𝑢))
Assertion
Ref Expression
opncldf2 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝐹𝐴) = (𝑋𝐴))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐽   𝑢,𝑋
Allowed substitution hint:   𝐹(𝑢)

Proof of Theorem opncldf2
StepHypRef Expression
1 opncldf.2 . 2 𝐹 = (𝑢𝐽 ↦ (𝑋𝑢))
2 difeq2 4112 . 2 (𝑢 = 𝐴 → (𝑋𝑢) = (𝑋𝐴))
3 simpr 483 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝐽)
4 opncldf.1 . . 3 𝑋 = 𝐽
54opncld 23025 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝑋𝐴) ∈ (Clsd‘𝐽))
61, 2, 3, 5fvmptd3 7024 1 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝐹𝐴) = (𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  cdif 3943   cuni 4905  cmpt 5228  cfv 6546  Topctop 22883  Clsdccld 23008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-iota 6498  df-fun 6548  df-fv 6554  df-top 22884  df-cld 23011
This theorem is referenced by: (None)
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