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Theorem opncldeqv 48799
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 2737 . . . 4 𝐽 = 𝐽
21cldopn 23039 . . 3 (𝑦 ∈ (Clsd‘𝐽) → ( 𝐽𝑦) ∈ 𝐽)
32adantl 481 . 2 ((𝜑𝑦 ∈ (Clsd‘𝐽)) → ( 𝐽𝑦) ∈ 𝐽)
4 opncldeqv.1 . . 3 (𝜑𝐽 ∈ Top)
51opncld 23041 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
6 elssuni 4937 . . . . . . . . 9 (𝑥𝐽𝑥 𝐽)
7 dfss4 4269 . . . . . . . . 9 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
86, 7sylib 218 . . . . . . . 8 (𝑥𝐽 → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
98eqcomd 2743 . . . . . . 7 (𝑥𝐽𝑥 = ( 𝐽 ∖ ( 𝐽𝑥)))
109adantl 481 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥𝐽) → 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥)))
115, 10jca 511 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥))))
12 eleq1 2829 . . . . . 6 (𝑦 = ( 𝐽𝑥) → (𝑦 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
13 difeq2 4120 . . . . . . 7 (𝑦 = ( 𝐽𝑥) → ( 𝐽𝑦) = ( 𝐽 ∖ ( 𝐽𝑥)))
1413eqeq2d 2748 . . . . . 6 (𝑦 = ( 𝐽𝑥) → (𝑥 = ( 𝐽𝑦) ↔ 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥))))
1512, 14anbi12d 632 . . . . 5 (𝑦 = ( 𝐽𝑥) → ((𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽𝑦)) ↔ (( 𝐽𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥)))))
165, 11, 15spcedv 3598 . . . 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 234 . . 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 5408 1 (𝜑 → (∀𝑥𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  cdif 3948  wss 3951   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-top 22900  df-cld 23027
This theorem is referenced by: (None)
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