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Theorem opncldf1 20698
 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 20647 . 2 ((𝐽 ∈ Top ∧ 𝑢𝐽) → (𝑋𝑢) ∈ (Clsd‘𝐽))
42cldopn 20645 . . 3 (𝑥 ∈ (Clsd‘𝐽) → (𝑋𝑥) ∈ 𝐽)
54adantl 481 . 2 ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋𝑥) ∈ 𝐽)
62cldss 20643 . . . . . . 7 (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋)
76ad2antll 761 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑥𝑋)
8 dfss4 3820 . . . . . 6 (𝑥𝑋 ↔ (𝑋 ∖ (𝑋𝑥)) = 𝑥)
97, 8sylib 207 . . . . 5 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋𝑥)) = 𝑥)
109eqcomd 2616 . . . 4 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑥 = (𝑋 ∖ (𝑋𝑥)))
11 difeq2 3684 . . . . 5 (𝑢 = (𝑋𝑥) → (𝑋𝑢) = (𝑋 ∖ (𝑋𝑥)))
1211eqeq2d 2620 . . . 4 (𝑢 = (𝑋𝑥) → (𝑥 = (𝑋𝑢) ↔ 𝑥 = (𝑋 ∖ (𝑋𝑥))))
1310, 12syl5ibrcom 236 . . 3 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋𝑥) → 𝑥 = (𝑋𝑢)))
142eltopss 20537 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢𝐽) → 𝑢𝑋)
1514adantrr 749 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑢𝑋)
16 dfss4 3820 . . . . . 6 (𝑢𝑋 ↔ (𝑋 ∖ (𝑋𝑢)) = 𝑢)
1715, 16sylib 207 . . . . 5 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋𝑢)) = 𝑢)
1817eqcomd 2616 . . . 4 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → 𝑢 = (𝑋 ∖ (𝑋𝑢)))
19 difeq2 3684 . . . . 5 (𝑥 = (𝑋𝑢) → (𝑋𝑥) = (𝑋 ∖ (𝑋𝑢)))
2019eqeq2d 2620 . . . 4 (𝑥 = (𝑋𝑢) → (𝑢 = (𝑋𝑥) ↔ 𝑢 = (𝑋 ∖ (𝑋𝑢))))
2118, 20syl5ibrcom 236 . . 3 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑥 = (𝑋𝑢) → 𝑢 = (𝑋𝑥)))
2213, 21impbid 201 . 2 ((𝐽 ∈ Top ∧ (𝑢𝐽𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋𝑥) ↔ 𝑥 = (𝑋𝑢)))
231, 3, 5, 22f1ocnv2d 6784 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 1475   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  ∪ cuni 4372   ↦ cmpt 4643  ◡ccnv 5037  –1-1-onto→wf1o 5803  ‘cfv 5804  Topctop 20517  Clsdccld 20630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633 This theorem is referenced by:  opncldf3  20700  cmpfi  21021
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