Theorem opncldeqv 46083

Description: Conditions on open sets are equivalent to conditions on closed sets. (Contributed by Zhi Wang, 30-Aug-2024.)

Hypotheses

Ref	Expression
opncldeqv.1	⊢ (𝜑 → 𝐽 ∈ Top)
opncldeqv.2	⊢ ((𝜑 ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
opncldeqv	⊢ (𝜑 → (∀𝑥 ∈ 𝐽 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

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldopn 22090	. . 3 ⊢ (𝑦 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ 𝐽)
3	2	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑦) ∈ 𝐽)
4		opncldeqv.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
5	1	opncld 22092	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
6		elssuni 4868	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
7		dfss4 4189	. . . . . . . . 9 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
8	6, 7	sylib 217	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
9	8	eqcomd 2744	. . . . . . 7 ⊢ (𝑥 ∈ 𝐽 → 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))
10	9	adantl 481	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))
11	5, 10	jca 511	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))))
12		eleq1 2826	. . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑦 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
13		difeq2 4047	. . . . . . 7 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (∪ 𝐽 ∖ 𝑦) = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))
14	13	eqeq2d 2749	. . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑥 = (∪ 𝐽 ∖ 𝑦) ↔ 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))))
15	12, 14	anbi12d 630	. . . . 5 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → ((𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) ↔ ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))))
16	5, 11, 15	spcedv 3527	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)))
17		df-rex 3069	. . . 4 ⊢ (∃𝑦 ∈ (Clsd‘𝐽)𝑥 = (∪ 𝐽 ∖ 𝑦) ↔ ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)))
18	16, 17	sylibr 233	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → ∃𝑦 ∈ (Clsd‘𝐽)𝑥 = (∪ 𝐽 ∖ 𝑦))
19	4, 18	sylan 579	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ∃𝑦 ∈ (Clsd‘𝐽)𝑥 = (∪ 𝐽 ∖ 𝑦))
20		opncldeqv.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) → (𝜓 ↔ 𝜒))
21	3, 19, 20	ralxfrd 5326	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ⊆ wss 3883  ∪ cuni 4836  ‘cfv 6418  Topctop 21950  Clsdccld 22075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-top 21951  df-cld 22078

This theorem is referenced by: (None)