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	⊢ (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 2737	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldopn 23039	. . 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 23041	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
6		elssuni 4937	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
7		dfss4 4269	. . . . . . . . 9 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
8	6, 7	sylib 218	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
9	8	eqcomd 2743	. . . . . . 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 2829	. . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑦 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
13		difeq2 4120	. . . . . . 7 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (∪ 𝐽 ∖ 𝑦) = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))
14	13	eqeq2d 2748	. . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑥 = (∪ 𝐽 ∖ 𝑦) ↔ 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))))
15	12, 14	anbi12d 632	. . . . 5 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → ((𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) ↔ ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))))
16	5, 11, 15	spcedv 3598	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)))
17		df-rex 3071	. . . 4 ⊢ (∃𝑦 ∈ (Clsd‘𝐽)𝑥 = (∪ 𝐽 ∖ 𝑦) ↔ ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)))
18	16, 17	sylibr 234	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → ∃𝑦 ∈ (Clsd‘𝐽)𝑥 = (∪ 𝐽 ∖ 𝑦))
19	4, 18	sylan 580	. 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 5408	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))

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)