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Theorem opncldf3 21786
 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 21726 . . . 4 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 opncldf.1 . . . . . 6 𝑋 = 𝐽
3 opncldf.2 . . . . . 6 𝐹 = (𝑢𝐽 ↦ (𝑋𝑢))
42, 3opncldf1 21784 . . . . 5 (𝐽 ∈ Top → (𝐹:𝐽1-1-onto→(Clsd‘𝐽) ∧ 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))))
54simprd 499 . . . 4 (𝐽 ∈ Top → 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)))
61, 5syl 17 . . 3 (𝐵 ∈ (Clsd‘𝐽) → 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)))
76fveq1d 6660 . 2 (𝐵 ∈ (Clsd‘𝐽) → (𝐹𝐵) = ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵))
82cldopn 21731 . . 3 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
9 difeq2 4022 . . . 4 (𝑥 = 𝐵 → (𝑋𝑥) = (𝑋𝐵))
10 eqid 2758 . . . 4 (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)) = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))
119, 10fvmptg 6757 . . 3 ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝑋𝐵) ∈ 𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵) = (𝑋𝐵))
128, 11mpdan 686 . 2 (𝐵 ∈ (Clsd‘𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵) = (𝑋𝐵))
137, 12eqtrd 2793 1 (𝐵 ∈ (Clsd‘𝐽) → (𝐹𝐵) = (𝑋𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ∖ cdif 3855  ∪ cuni 4798   ↦ cmpt 5112  ◡ccnv 5523  –1-1-onto→wf1o 6334  ‘cfv 6335  Topctop 21593  Clsdccld 21716 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-top 21594  df-cld 21719 This theorem is referenced by: (None)
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