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Theorem opncldf3 23109
Description: The values of the converse/inverse 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
opncldf3 (𝐵 ∈ (Clsd‘𝐽) → (𝐹𝐵) = (𝑋𝐵))
Distinct variable groups:   𝑢,𝐽   𝑢,𝑋
Allowed substitution hints:   𝐵(𝑢)   𝐹(𝑢)

Proof of Theorem opncldf3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cldrcl 23049 . . . 4 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 opncldf.1 . . . . . 6 𝑋 = 𝐽
3 opncldf.2 . . . . . 6 𝐹 = (𝑢𝐽 ↦ (𝑋𝑢))
42, 3opncldf1 23107 . . . . 5 (𝐽 ∈ Top → (𝐹:𝐽1-1-onto→(Clsd‘𝐽) ∧ 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))))
54simprd 495 . . . 4 (𝐽 ∈ Top → 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)))
61, 5syl 17 . . 3 (𝐵 ∈ (Clsd‘𝐽) → 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)))
76fveq1d 6908 . 2 (𝐵 ∈ (Clsd‘𝐽) → (𝐹𝐵) = ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵))
82cldopn 23054 . . 3 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
9 difeq2 4129 . . . 4 (𝑥 = 𝐵 → (𝑋𝑥) = (𝑋𝐵))
10 eqid 2734 . . . 4 (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)) = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))
119, 10fvmptg 7013 . . 3 ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝑋𝐵) ∈ 𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵) = (𝑋𝐵))
128, 11mpdan 687 . 2 (𝐵 ∈ (Clsd‘𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵) = (𝑋𝐵))
137, 12eqtrd 2774 1 (𝐵 ∈ (Clsd‘𝐽) → (𝐹𝐵) = (𝑋𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  cdif 3959   cuni 4911  cmpt 5230  ccnv 5687  1-1-ontowf1o 6561  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  ax-un 7753
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-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-top 22915  df-cld 23042
This theorem is referenced by: (None)
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