Theorem isclo2 23111

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 23110	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
3		eleq1w 2821	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
4	3	bibi2d 342	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
5	4	cbvralvw 3234	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ ∀𝑤 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
6	5	anbi2i 623	. . . . . . . 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		pm4.24 563	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
8		raaanv 4523	. . . . . . . 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 303	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
10		bibi1 351	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
11	10	biimpa 476	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
12	11	biimpcd 249	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → 𝑤 ∈ 𝐴))
13	12	ralimdv 3166	. . . . . . . . . 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 3983	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑤 ∈ 𝑦 𝑤 ∈ 𝐴)
16	14, 15	imbitrrdi 252	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))
17	16	ralimi 3080	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))
18	9, 17	sylbi 217	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))
19		eleq1w 2821	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
20	19	imbi1d 341	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
21	20	rspcv 3617	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → (𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
22		dfss3 3983	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴)
23	22	imbi2i 336	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝐴 → ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴))
24		r19.21v 3177	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴))
25	23, 24	bitr4i 278	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴))
26	21, 25	imbitrdi 251	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴)))
27		ssel 3988	. . . . . . . . . . 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 3166	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴)))
31	26, 30	jcad 512	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴))))
32		ralbiim 3110	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴)))
33	31, 32	imbitrrdi 252	. . . . . 6 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
34	18, 33	impbid2 226	. . . . 5 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
35	34	pm5.32i 574	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
36	35	rexbii 3091	. . 3 ⊢ (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
37	36	ralbii 3090	. 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	bitrdi 287	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 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067   ∩ cin 3961   ⊆ wss 3962  ∪ cuni 4911  ‘cfv 6562  Topctop 22914  Clsdccld 23039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-topgen 17489  df-top 22915  df-cld 23042

This theorem is referenced by:  connpconn  35219