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Theorem opncldf1 20869
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 20818 . 2 ((𝐽 ∈ Top ∧ 𝑢𝐽) → (𝑋𝑢) ∈ (Clsd‘𝐽))
42cldopn 20816 . . 3 (𝑥 ∈ (Clsd‘𝐽) → (𝑋𝑥) ∈ 𝐽)
54adantl 482 . 2 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋𝑥) ∈ 𝐽)
62cldss 20814 . . . . . . 7 (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋)
76ad2antll 764 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑥𝑋)
8 dfss4 3850 . . . . . 6 (𝑥𝑋 ↔ (𝑋 ∖ (𝑋𝑥)) = 𝑥)
97, 8sylib 208 . . . . 5 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋𝑥)) = 𝑥)
109eqcomd 2626 . . . 4 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑥 = (𝑋 ∖ (𝑋𝑥)))
11 difeq2 3714 . . . . 5 (𝑢 = (𝑋𝑥) → (𝑋𝑢) = (𝑋 ∖ (𝑋𝑥)))
1211eqeq2d 2630 . . . 4 (𝑢 = (𝑋𝑥) → (𝑥 = (𝑋𝑢) ↔ 𝑥 = (𝑋 ∖ (𝑋𝑥))))
1310, 12syl5ibrcom 237 . . 3 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋𝑥) → 𝑥 = (𝑋𝑢)))
142eltopss 20693 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢𝐽) → 𝑢𝑋)
1514adantrr 752 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑢𝑋)
16 dfss4 3850 . . . . . 6 (𝑢𝑋 ↔ (𝑋 ∖ (𝑋𝑢)) = 𝑢)
1715, 16sylib 208 . . . . 5 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋𝑢)) = 𝑢)
1817eqcomd 2626 . . . 4 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑢 = (𝑋 ∖ (𝑋𝑢)))
19 difeq2 3714 . . . . 5 (𝑥 = (𝑋𝑢) → (𝑋𝑥) = (𝑋 ∖ (𝑋𝑢)))
2019eqeq2d 2630 . . . 4 (𝑥 = (𝑋𝑢) → (𝑢 = (𝑋𝑥) ↔ 𝑢 = (𝑋 ∖ (𝑋𝑢))))
2118, 20syl5ibrcom 237 . . 3 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑥 = (𝑋𝑢) → 𝑢 = (𝑋𝑥)))
2213, 21impbid 202 . 2 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋𝑥) ↔ 𝑥 = (𝑋𝑢)))
231, 3, 5, 22f1ocnv2d 6871 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 384   = wceq 1481  wcel 1988  cdif 3564  wss 3567   cuni 4427  cmpt 4720  ccnv 5103  1-1-ontowf1o 5875  cfv 5876  Topctop 20679  Clsdccld 20801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-top 20680  df-cld 20804
This theorem is referenced by:  opncldf3  20871  cmpfi  21192
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