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Theorem opncldf3 23094
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 23034 . . . 4 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 opncldf.1 . . . . . 6 𝑋 = 𝐽
3 opncldf.2 . . . . . 6 𝐹 = (𝑢𝐽 ↦ (𝑋𝑢))
42, 3opncldf1 23092 . . . . 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 23039 . . 3 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
9 difeq2 4120 . . . 4 (𝑥 = 𝐵 → (𝑋𝑥) = (𝑋𝐵))
10 eqid 2737 . . . 4 (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥)) = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))
119, 10fvmptg 7014 . . 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 2777 1 (𝐵 ∈ (Clsd‘𝐽) → (𝐹𝐵) = (𝑋𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cdif 3948   cuni 4907  cmpt 5225  ccnv 5684  1-1-ontowf1o 6560  cfv 6561  Topctop 22899  Clsdccld 23024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027
This theorem is referenced by: (None)
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