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Theorem opncldeqv 45701
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 21775 . . 3 (𝑦 ∈ (Clsd‘𝐽) → ( 𝐽𝑦) ∈ 𝐽)
32adantl 485 . 2 ((𝜑𝑦 ∈ (Clsd‘𝐽)) → ( 𝐽𝑦) ∈ 𝐽)
4 opncldeqv.1 . . 3 (𝜑𝐽 ∈ Top)
51opncld 21777 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
6 elssuni 4825 . . . . . . . . 9 (𝑥𝐽𝑥 𝐽)
7 dfss4 4147 . . . . . . . . 9 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
86, 7sylib 221 . . . . . . . 8 (𝑥𝐽 → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
98eqcomd 2744 . . . . . . 7 (𝑥𝐽𝑥 = ( 𝐽 ∖ ( 𝐽𝑥)))
109adantl 485 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥𝐽) → 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥)))
115, 10jca 515 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥))))
12 eleq1 2820 . . . . . 6 (𝑦 = ( 𝐽𝑥) → (𝑦 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
13 difeq2 4005 . . . . . . 7 (𝑦 = ( 𝐽𝑥) → ( 𝐽𝑦) = ( 𝐽 ∖ ( 𝐽𝑥)))
1413eqeq2d 2749 . . . . . 6 (𝑦 = ( 𝐽𝑥) → (𝑥 = ( 𝐽𝑦) ↔ 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥))))
1512, 14anbi12d 634 . . . . 5 (𝑦 = ( 𝐽𝑥) → ((𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽𝑦)) ↔ (( 𝐽𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽 ∖ ( 𝐽𝑥)))))
165, 11, 15spcedv 3500 . . . 4 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽𝑦)))
17 df-rex 3059 . . . 4 (∃𝑦 ∈ (Clsd‘𝐽)𝑥 = ( 𝐽𝑦) ↔ ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = ( 𝐽𝑦)))
1816, 17sylibr 237 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ∃𝑦 ∈ (Clsd‘𝐽)𝑥 = ( 𝐽𝑦))
194, 18sylan 583 . 2 ((𝜑𝑥𝐽) → ∃𝑦 ∈ (Clsd‘𝐽)𝑥 = ( 𝐽𝑦))
20 opncldeqv.2 . 2 ((𝜑𝑥 = ( 𝐽𝑦)) → (𝜓𝜒))
213, 19, 20ralxfrd 5272 1 (𝜑 → (∀𝑥𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2113  wral 3053  wrex 3054  cdif 3838  wss 3841   cuni 4793  cfv 6333  Topctop 21637  Clsdccld 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6291  df-fun 6335  df-fn 6336  df-fv 6341  df-top 21638  df-cld 21763
This theorem is referenced by: (None)
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