Theorem opncldeqv 48698

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 2735	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldopn 23055	. . 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 23057	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
6		elssuni 4942	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
7		dfss4 4275	. . . . . . . . 9 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
8	6, 7	sylib 218	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
9	8	eqcomd 2741	. . . . . . 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 2827	. . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑦 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
13		difeq2 4130	. . . . . . 7 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (∪ 𝐽 ∖ 𝑦) = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))
14	13	eqeq2d 2746	. . . . . 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 3069	. . . 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 5414	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ∖ cdif 3960   ⊆ wss 3963  ∪ cuni 4912  ‘cfv 6563  Topctop 22915  Clsdccld 23040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-top 22916  df-cld 23043

This theorem is referenced by: (None)