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Theorem opncldeqv 47524
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 2732 . . . 4 𝐽 = 𝐽
21cldopn 22534 . . 3 (𝑦 ∈ (Clsd‘𝐽) → ( 𝐽𝑦) ∈ 𝐽)
32adantl 482 . 2 ((𝜑𝑦 ∈ (Clsd‘𝐽)) → ( 𝐽𝑦) ∈ 𝐽)
4 opncldeqv.1 . . 3 (𝜑𝐽 ∈ Top)
51opncld 22536 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
6 elssuni 4941 . . . . . . . . 9 (𝑥𝐽𝑥 𝐽)
7 dfss4 4258 . . . . . . . . 9 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
86, 7sylib 217 . . . . . . . 8 (𝑥𝐽 → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
98eqcomd 2738 . . . . . . 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 2821 . . . . . 6 (𝑦 = ( 𝐽𝑥) → (𝑦 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
13 difeq2 4116 . . . . . . 7 (𝑦 = ( 𝐽𝑥) → ( 𝐽𝑦) = ( 𝐽 ∖ ( 𝐽𝑥)))
1413eqeq2d 2743 . . . . . 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 3588 . . . 4 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽𝑦)))
17 df-rex 3071 . . . 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 5406 1 (𝜑 → (∀𝑥𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wral 3061  wrex 3070  cdif 3945  wss 3948   cuni 4908  cfv 6543  Topctop 22394  Clsdccld 22519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-top 22395  df-cld 22522
This theorem is referenced by: (None)
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