Theorem isclo2 20940

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 20939	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
3		eleq1 2718	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
4	3	bibi2d 331	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
5	4	cbvralv 3201	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ ∀𝑤 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
6	5	anbi2i 730	. . . . . . . 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		pm4.24 676	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
8		raaanv 4116	. . . . . . . 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 340	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
11	10	biimpa 500	. . . . . . . . . . . 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 2992	. . . . . . . . . 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 3625	. . . . . . . . 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 2981	. . . . . . 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 2718	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
20	19	imbi1d 330	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
21	20	rspcv 3336	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → (𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
22		dfss3 3625	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴)
23	22	imbi2i 325	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝐴 → ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴))
24		r19.21v 2989	. . . . . . . . . 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 3630	. . . . . . . . . . 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 2992	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴)))
31	26, 30	jcad 554	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴))))
32		ralbiim 3098	. . . . . . 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 670	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
36	35	rexbii 3070	. . 3 ⊢ (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
37	36	ralbii 3009	. 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 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ∩ cin 3606   ⊆ wss 3607  ∪ cuni 4468  ‘cfv 5926  Topctop 20746  Clsdccld 20868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151  df-top 20747  df-cld 20871

This theorem is referenced by:  connpconn  31343