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Theorem opncldf2 23108
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 4129 . 2 (𝑢 = 𝐴 → (𝑋𝑢) = (𝑋𝐴))
3 simpr 484 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝐽)
4 opncldf.1 . . 3 𝑋 = 𝐽
54opncld 23056 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝑋𝐴) ∈ (Clsd‘𝐽))
61, 2, 3, 5fvmptd3 7038 1 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝐹𝐴) = (𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  cdif 3959   cuni 4911  cmpt 5230  cfv 6562  Topctop 22914  Clsdccld 23039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-top 22915  df-cld 23042
This theorem is referenced by: (None)
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