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Theorem opncldf1 23092
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 23041 . 2 ((𝐽 ∈ Top ∧ 𝑢𝐽) → (𝑋𝑢) ∈ (Clsd‘𝐽))
42cldopn 23039 . . 3 (𝑥 ∈ (Clsd‘𝐽) → (𝑋𝑥) ∈ 𝐽)
54adantl 481 . 2 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋𝑥) ∈ 𝐽)
62cldss 23037 . . . . . . 7 (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋)
76ad2antll 729 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑥𝑋)
8 dfss4 4269 . . . . . 6 (𝑥𝑋 ↔ (𝑋 ∖ (𝑋𝑥)) = 𝑥)
97, 8sylib 218 . . . . 5 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋𝑥)) = 𝑥)
109eqcomd 2743 . . . 4 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑥 = (𝑋 ∖ (𝑋𝑥)))
11 difeq2 4120 . . . . 5 (𝑢 = (𝑋𝑥) → (𝑋𝑢) = (𝑋 ∖ (𝑋𝑥)))
1211eqeq2d 2748 . . . 4 (𝑢 = (𝑋𝑥) → (𝑥 = (𝑋𝑢) ↔ 𝑥 = (𝑋 ∖ (𝑋𝑥))))
1310, 12syl5ibrcom 247 . . 3 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋𝑥) → 𝑥 = (𝑋𝑢)))
142eltopss 22913 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢𝐽) → 𝑢𝑋)
1514adantrr 717 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑢𝑋)
16 dfss4 4269 . . . . . 6 (𝑢𝑋 ↔ (𝑋 ∖ (𝑋𝑢)) = 𝑢)
1715, 16sylib 218 . . . . 5 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋𝑢)) = 𝑢)
1817eqcomd 2743 . . . 4 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑢 = (𝑋 ∖ (𝑋𝑢)))
19 difeq2 4120 . . . . 5 (𝑥 = (𝑋𝑢) → (𝑋𝑥) = (𝑋 ∖ (𝑋𝑢)))
2019eqeq2d 2748 . . . 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 7686 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 1540  wcel 2108  cdif 3948  wss 3951   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:  opncldf3  23094  cmpfi  23416
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