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Theorem opncldf3 23034
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 22974 . . . 4 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 opncldf.1 . . . . . 6 𝑋 = 𝐽
3 opncldf.2 . . . . . 6 𝐹 = (𝑢𝐽 ↦ (𝑋𝑢))
42, 3opncldf1 23032 . . . . 5 (𝐽 ∈ Top → (𝐹:𝐽1-1-onto→(Clsd‘𝐽) ∧ 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))))
54simprd 494 . . . 4 (𝐽 ∈ Top → 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)))
61, 5syl 17 . . 3 (𝐵 ∈ (Clsd‘𝐽) → 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)))
76fveq1d 6898 . 2 (𝐵 ∈ (Clsd‘𝐽) → (𝐹𝐵) = ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵))
82cldopn 22979 . . 3 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
9 difeq2 4112 . . . 4 (𝑥 = 𝐵 → (𝑋𝑥) = (𝑋𝐵))
10 eqid 2725 . . . 4 (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)) = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))
119, 10fvmptg 7002 . . 3 ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝑋𝐵) ∈ 𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵) = (𝑋𝐵))
128, 11mpdan 685 . 2 (𝐵 ∈ (Clsd‘𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵) = (𝑋𝐵))
137, 12eqtrd 2765 1 (𝐵 ∈ (Clsd‘𝐽) → (𝐹𝐵) = (𝑋𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cdif 3941   cuni 4909  cmpt 5232  ccnv 5677  1-1-ontowf1o 6548  cfv 6549  Topctop 22839  Clsdccld 22964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-top 22840  df-cld 22967
This theorem is referenced by: (None)
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