Theorem isclo2 23022

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 23021	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
3		eleq1w 2808	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
4	3	bibi2d 341	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
5	4	cbvralvw 3225	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ ∀𝑤 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
6	5	anbi2i 621	. . . . . . . 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		pm4.24 562	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
8		raaanv 4522	. . . . . . . 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 302	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
10		bibi1 350	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴) ↔ (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)))
11	10	biimpa 475	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
12	11	biimpcd 248	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → 𝑤 ∈ 𝐴))
13	12	ralimdv 3159	. . . . . . . . . 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 3966	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑤 ∈ 𝑦 𝑤 ∈ 𝐴)
16	14, 15	imbitrrdi 251	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))
17	16	ralimi 3073	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) → ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))
18	9, 17	sylbi 216	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴))
19		eleq1w 2808	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
20	19	imbi1d 340	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
21	20	rspcv 3603	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → (𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
22		dfss3 3966	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴)
23	22	imbi2i 335	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝐴 → ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴))
24		r19.21v 3170	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴))
25	23, 24	bitr4i 277	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 → 𝑦 ⊆ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴))
26	21, 25	imbitrdi 250	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴)))
27		ssel 3971	. . . . . . . . . . 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 3159	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴)))
31	26, 30	jcad 511	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴))))
32		ralbiim 3103	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 → 𝑧 ∈ 𝐴) ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑥 ∈ 𝐴)))
33	31, 32	imbitrrdi 251	. . . . . 6 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴) → ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
34	18, 33	impbid2 225	. . . . 5 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
35	34	pm5.32i 573	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
36	35	rexbii 3084	. . 3 ⊢ (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ∈ 𝐴 → 𝑦 ⊆ 𝐴)))
37	36	ralbii 3083	. 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 286	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 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3051  ∃wrex 3060   ∩ cin 3944   ⊆ wss 3945  ∪ cuni 4908  ‘cfv 6547  Topctop 22825  Clsdccld 22950

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 2696  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739

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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6499  df-fun 6549  df-fv 6555  df-topgen 17424  df-top 22826  df-cld 22953

This theorem is referenced by:  connpconn  34915