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Theorem opncldf1 23107
Description: A bijection useful for converting statements about open sets to statements about closed sets and vice versa. (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
opncldf1 (𝐽 ∈ Top → (𝐹:𝐽1-1-onto→(Clsd‘𝐽) ∧ 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑢,𝐽   𝑢,𝑋,𝑥
Allowed substitution hint:   𝐹(𝑢)

Proof of Theorem opncldf1
StepHypRef Expression
1 opncldf.2 . 2 𝐹 = (𝑢𝐽 ↦ (𝑋𝑢))
2 opncldf.1 . . 3 𝑋 = 𝐽
32opncld 23056 . 2 ((𝐽 ∈ Top ∧ 𝑢𝐽) → (𝑋𝑢) ∈ (Clsd‘𝐽))
42cldopn 23054 . . 3 (𝑥 ∈ (Clsd‘𝐽) → (𝑋𝑥) ∈ 𝐽)
54adantl 481 . 2 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋𝑥) ∈ 𝐽)
62cldss 23052 . . . . . . 7 (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋)
76ad2antll 729 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑥𝑋)
8 dfss4 4274 . . . . . 6 (𝑥𝑋 ↔ (𝑋 ∖ (𝑋𝑥)) = 𝑥)
97, 8sylib 218 . . . . 5 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋𝑥)) = 𝑥)
109eqcomd 2740 . . . 4 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑥 = (𝑋 ∖ (𝑋𝑥)))
11 difeq2 4129 . . . . 5 (𝑢 = (𝑋𝑥) → (𝑋𝑢) = (𝑋 ∖ (𝑋𝑥)))
1211eqeq2d 2745 . . . 4 (𝑢 = (𝑋𝑥) → (𝑥 = (𝑋𝑢) ↔ 𝑥 = (𝑋 ∖ (𝑋𝑥))))
1310, 12syl5ibrcom 247 . . 3 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋𝑥) → 𝑥 = (𝑋𝑢)))
142eltopss 22928 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢𝐽) → 𝑢𝑋)
1514adantrr 717 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑢𝑋)
16 dfss4 4274 . . . . . 6 (𝑢𝑋 ↔ (𝑋 ∖ (𝑋𝑢)) = 𝑢)
1715, 16sylib 218 . . . . 5 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋𝑢)) = 𝑢)
1817eqcomd 2740 . . . 4 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑢 = (𝑋 ∖ (𝑋𝑢)))
19 difeq2 4129 . . . . 5 (𝑥 = (𝑋𝑢) → (𝑋𝑥) = (𝑋 ∖ (𝑋𝑢)))
2019eqeq2d 2745 . . . 4 (𝑥 = (𝑋𝑢) → (𝑢 = (𝑋𝑥) ↔ 𝑢 = (𝑋 ∖ (𝑋𝑢))))
2118, 20syl5ibrcom 247 . . 3 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑥 = (𝑋𝑢) → 𝑢 = (𝑋𝑥)))
2213, 21impbid 212 . 2 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋𝑥) ↔ 𝑥 = (𝑋𝑢)))
231, 3, 5, 22f1ocnv2d 7685 1 (𝐽 ∈ Top → (𝐹:𝐽1-1-onto→(Clsd‘𝐽) ∧ 𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  cdif 3959  wss 3962   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:  opncldf3  23109  cmpfi  23431
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