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Theorem opncldf2 21688
 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 4068 . 2 (𝑢 = 𝐴 → (𝑋𝑢) = (𝑋𝐴))
3 simpr 488 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝐽)
4 opncldf.1 . . 3 𝑋 = 𝐽
54opncld 21636 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝑋𝐴) ∈ (Clsd‘𝐽))
61, 2, 3, 5fvmptd3 6773 1 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝐹𝐴) = (𝑋𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114   ∖ cdif 3905  ∪ cuni 4813   ↦ cmpt 5122  ‘cfv 6334  Topctop 21496  Clsdccld 21619 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-top 21497  df-cld 21622 This theorem is referenced by: (None)
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