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Theorem opncldf1 20936
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 20885 . 2 ((𝐽 ∈ Top ∧ 𝑢𝐽) → (𝑋𝑢) ∈ (Clsd‘𝐽))
42cldopn 20883 . . 3 (𝑥 ∈ (Clsd‘𝐽) → (𝑋𝑥) ∈ 𝐽)
54adantl 481 . 2 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋𝑥) ∈ 𝐽)
62cldss 20881 . . . . . . 7 (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋)
76ad2antll 765 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑥𝑋)
8 dfss4 3891 . . . . . 6 (𝑥𝑋 ↔ (𝑋 ∖ (𝑋𝑥)) = 𝑥)
97, 8sylib 208 . . . . 5 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋𝑥)) = 𝑥)
109eqcomd 2657 . . . 4 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑥 = (𝑋 ∖ (𝑋𝑥)))
11 difeq2 3755 . . . . 5 (𝑢 = (𝑋𝑥) → (𝑋𝑢) = (𝑋 ∖ (𝑋𝑥)))
1211eqeq2d 2661 . . . 4 (𝑢 = (𝑋𝑥) → (𝑥 = (𝑋𝑢) ↔ 𝑥 = (𝑋 ∖ (𝑋𝑥))))
1310, 12syl5ibrcom 237 . . 3 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋𝑥) → 𝑥 = (𝑋𝑢)))
142eltopss 20760 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢𝐽) → 𝑢𝑋)
1514adantrr 753 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑢𝑋)
16 dfss4 3891 . . . . . 6 (𝑢𝑋 ↔ (𝑋 ∖ (𝑋𝑢)) = 𝑢)
1715, 16sylib 208 . . . . 5 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋𝑢)) = 𝑢)
1817eqcomd 2657 . . . 4 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑢 = (𝑋 ∖ (𝑋𝑢)))
19 difeq2 3755 . . . . 5 (𝑥 = (𝑋𝑢) → (𝑋𝑥) = (𝑋 ∖ (𝑋𝑢)))
2019eqeq2d 2661 . . . 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 6928 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 383   = wceq 1523  wcel 2030  cdif 3604  wss 3607   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:  opncldf3  20938  cmpfi  21259
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