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Theorem opncldeqv 46195
Description: Conditions on open sets are equivalent to conditions on closed sets. (Contributed by Zhi Wang, 30-Aug-2024.)
Hypotheses
Ref Expression
opncldeqv.1 (𝜑𝐽 ∈ Top)
opncldeqv.2 ((𝜑𝑥 = ( 𝐽𝑦)) → (𝜓𝜒))
Assertion
Ref Expression
opncldeqv (𝜑 → (∀𝑥𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))
Distinct variable groups:   𝑥,𝐽,𝑦   𝜒,𝑥   𝜑,𝑥,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem opncldeqv
StepHypRef Expression
1 eqid 2738 . . . 4 𝐽 = 𝐽
21cldopn 22182 . . 3 (𝑦 ∈ (Clsd‘𝐽) → ( 𝐽𝑦) ∈ 𝐽)
32adantl 482 . 2 ((𝜑𝑦 ∈ (Clsd‘𝐽)) → ( 𝐽𝑦) ∈ 𝐽)
4 opncldeqv.1 . . 3 (𝜑𝐽 ∈ Top)
51opncld 22184 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
6 elssuni 4871 . . . . . . . . 9 (𝑥𝐽𝑥 𝐽)
7 dfss4 4192 . . . . . . . . 9 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
86, 7sylib 217 . . . . . . . 8 (𝑥𝐽 → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
98eqcomd 2744 . . . . . . 7 (𝑥𝐽𝑥 = ( 𝐽 ∖ ( 𝐽𝑥)))
109adantl 482 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥𝐽) → 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥)))
115, 10jca 512 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥))))
12 eleq1 2826 . . . . . 6 (𝑦 = ( 𝐽𝑥) → (𝑦 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
13 difeq2 4051 . . . . . . 7 (𝑦 = ( 𝐽𝑥) → ( 𝐽𝑦) = ( 𝐽 ∖ ( 𝐽𝑥)))
1413eqeq2d 2749 . . . . . 6 (𝑦 = ( 𝐽𝑥) → (𝑥 = ( 𝐽𝑦) ↔ 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥))))
1512, 14anbi12d 631 . . . . 5 (𝑦 = ( 𝐽𝑥) → ((𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽𝑦)) ↔ (( 𝐽𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥)))))
165, 11, 15spcedv 3537 . . . 4 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽𝑦)))
17 df-rex 3070 . . . 4 (∃𝑦 ∈ (Clsd‘𝐽)𝑥 = ( 𝐽𝑦) ↔ ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽𝑦)))
1816, 17sylibr 233 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ∃𝑦 ∈ (Clsd‘𝐽)𝑥 = ( 𝐽𝑦))
194, 18sylan 580 . 2 ((𝜑𝑥𝐽) → ∃𝑦 ∈ (Clsd‘𝐽)𝑥 = ( 𝐽𝑦))
20 opncldeqv.2 . 2 ((𝜑𝑥 = ( 𝐽𝑦)) → (𝜓𝜒))
213, 19, 20ralxfrd 5331 1 (𝜑 → (∀𝑥𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wex 1782  wcel 2106  wral 3064  wrex 3065  cdif 3884  wss 3887   cuni 4839  cfv 6433  Topctop 22042  Clsdccld 22167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-top 22043  df-cld 22170
This theorem is referenced by: (None)
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