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Theorem opncldf3 20938
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 20878 . . . 4 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 opncldf.1 . . . . . 6 𝑋 = 𝐽
3 opncldf.2 . . . . . 6 𝐹 = (𝑢𝐽 ↦ (𝑋𝑢))
42, 3opncldf1 20936 . . . . 5 (𝐽 ∈ Top → (𝐹:𝐽1-1-onto→(Clsd‘𝐽) ∧ 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))))
54simprd 478 . . . 4 (𝐽 ∈ Top → 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)))
61, 5syl 17 . . 3 (𝐵 ∈ (Clsd‘𝐽) → 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)))
76fveq1d 6231 . 2 (𝐵 ∈ (Clsd‘𝐽) → (𝐹𝐵) = ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵))
82cldopn 20883 . . 3 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
9 difeq2 3755 . . . 4 (𝑥 = 𝐵 → (𝑋𝑥) = (𝑋𝐵))
10 eqid 2651 . . . 4 (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)) = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))
119, 10fvmptg 6319 . . 3 ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝑋𝐵) ∈ 𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵) = (𝑋𝐵))
128, 11mpdan 703 . 2 (𝐵 ∈ (Clsd‘𝐽) → ((𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))‘𝐵) = (𝑋𝐵))
137, 12eqtrd 2685 1 (𝐵 ∈ (Clsd‘𝐽) → (𝐹𝐵) = (𝑋𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cdif 3604   cuni 4468  cmpt 4762  ccnv 5142  1-1-ontowf1o 5925  cfv 5926  Topctop 20746  Clsdccld 20868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-top 20747  df-cld 20871
This theorem is referenced by: (None)
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