Theorem opncldeqv 47998

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 2728	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldopn 22955	. . 3 ⊢ (𝑦 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ 𝐽)
3	2	adantl 480	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑦) ∈ 𝐽)
4		opncldeqv.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
5	1	opncld 22957	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
6		elssuni 4944	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
7		dfss4 4261	. . . . . . . . 9 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
8	6, 7	sylib 217	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐽 → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
9	8	eqcomd 2734	. . . . . . 7 ⊢ (𝑥 ∈ 𝐽 → 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))
10	9	adantl 480	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))
11	5, 10	jca 510	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))))
12		eleq1 2817	. . . . . 6 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑦 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
13		difeq2 4116	. . . . . . 7 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (∪ 𝐽 ∖ 𝑦) = (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))
14	13	eqeq2d 2739	. . . . . 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 3587	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → ∃𝑦(𝑦 ∈ (Clsd‘𝐽) ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)))
17		df-rex 3068	. . . 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 578	. 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 5412	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∀wral 3058  ∃wrex 3067   ∖ cdif 3946   ⊆ wss 3949  ∪ cuni 4912  ‘cfv 6553  Topctop 22815  Clsdccld 22940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fn 6556  df-fv 6561  df-top 22816  df-cld 22943

This theorem is referenced by: (None)