Theorem isclo2 20797

Description: A set 𝐴 is clopen iff for every point 𝑥 in the space there is a neighborhood 𝑦 of 𝑥 which is either disjoint from 𝐴 or contained in 𝐴. (Contributed by Mario Carneiro, 7-Jul-2015.)

Hypothesis

Ref	Expression
isclo.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
isclo2	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐽,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem isclo2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isclo.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	isclo 20796	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
3		eleq1 2692	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
4	3	bibi2d 332	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
5	4	cbvralv 3164	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ ∀𝑤 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
6	5	anbi2i 729	. . . . . . . 8 ⊢ ((∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ ∀𝑤 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
7		pm4.24 674	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
8		raaanv 4060	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) ↔ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ ∀𝑤 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
9	6, 7, 8	3bitr4i 292	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
10		bibi1 341	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
11	10	biimpa 501	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
12	11	biimpcd 239	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → 𝑤 ∈ 𝐴))
13	12	ralimdv 2962	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → (∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → ∀𝑤 ∈ 𝑦 𝑤 ∈ 𝐴))
14	13	com12 32	. . . . . . . . 9 ⊢ (∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → (𝑧 ∈ 𝐴 → ∀𝑤 ∈ 𝑦 𝑤 ∈ 𝐴))
15		dfss3 3578	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑤 ∈ 𝑦 𝑤 ∈ 𝐴)
16	14, 15	syl6ibr 242	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))
17	16	ralimi 2952	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))
18	9, 17	sylbi 207	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))
19		eleq1 2692	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
20	19	imbi1d 331	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
21	20	rspcv 3296	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → (𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
22		dfss3 3578	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴)
23	22	imbi2i 326	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝐴 → ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴))
24		r19.21v 2959	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴))
25	23, 24	bitr4i 267	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴))
26	21, 25	syl6ib 241	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴)))
27		ssel 3582	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝐴))
28	27	com12 32	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 → (𝑦 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
29	28	imim2d 57	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑦 → ((𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴)))
30	29	ralimdv 2962	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴)))
31	26, 30	jcad 555	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴))))
32		ralbiim 3067	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴)))
33	31, 32	syl6ibr 242	. . . . . 6 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
34	18, 33	impbid2 216	. . . . 5 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
35	34	pm5.32i 668	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
36	35	rexbii 3039	. . 3 ⊢ (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
37	36	ralbii 2979	. 2 ⊢ (∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
38	2, 37	syl6bb 276	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1992  ∀wral 2912  ∃wrex 2913   ∩ cin 3559   ⊆ wss 3560  ∪ cuni 4407  ‘cfv 5850  Topctop 20612  Clsdccld 20725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-topgen 16020  df-top 20616  df-cld 20728

This theorem is referenced by:  connpconn  30917