Theorem isclo 20701

Description: A set 𝐴 is clopen iff for every point 𝑥 in the space there is a neighborhood 𝑦 such that all the points in 𝑦 are in 𝐴 iff 𝑥 is. (Contributed by Mario Carneiro, 10-Mar-2015.)

Hypothesis

Ref	Expression
isclo.1	⊢ 𝑋 = ∪ 𝐽

Assertion

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

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

Proof of Theorem isclo

Step	Hyp	Ref	Expression
1		elin 3758	. 2 ⊢ (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ (𝐴 ∈ 𝐽 ∧ 𝐴 ∈ (Clsd‘𝐽)))
2		isclo.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld2 20642	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝐴) ∈ 𝐽))
4	3	anbi2d 736	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐴 ∈ 𝐽 ∧ 𝐴 ∈ (Clsd‘𝐽)) ↔ (𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐴) ∈ 𝐽)))
5		eltop2 20590	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
6		dfss3 3558	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴)
7		pm5.501 355	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
8	7	ralbidv 2969	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴 ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
9	6, 8	syl5bb 271	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
10	9	anbi2d 736	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
11	10	rexbidv 3034	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
12	11	ralbiia 2962	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
13	5, 12	syl6bb 275	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
14		eltop2 20590	. . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴))))
15		dfss3 3558	. . . . . . . . . 10 ⊢ (𝑦 ⊆ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ (𝑋 ∖ 𝐴))
16		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑦)
17		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐽)
18		elunii 4377	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐽) → 𝑧 ∈ ∪ 𝐽)
19	16, 17, 18	syl2anr 494	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ ∪ 𝐽)
20	19, 2	syl6eleqr 2699	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑋)
21		eldif 3550	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑋 ∖ 𝐴) ↔ (𝑧 ∈ 𝑋 ∧ ¬ 𝑧 ∈ 𝐴))
22	21	baib 942	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑋 → (𝑧 ∈ (𝑋 ∖ 𝐴) ↔ ¬ 𝑧 ∈ 𝐴))
23	20, 22	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → (𝑧 ∈ (𝑋 ∖ 𝐴) ↔ ¬ 𝑧 ∈ 𝐴))
24		eldifn 3695	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
25		nbn2 359	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑥 ∈ 𝐴 → (¬ 𝑧 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
26	24, 25	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) → (¬ 𝑧 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
27	26	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → (¬ 𝑧 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
28	23, 27	bitrd 267	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → (𝑧 ∈ (𝑋 ∖ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
29	28	ralbidva 2968	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → (∀𝑧 ∈ 𝑦 𝑧 ∈ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
30	15, 29	syl5bb 271	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → (𝑦 ⊆ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
31	30	anbi2d 736	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
32	31	rexbidva 3031	. . . . . . 7 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) → (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴)) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
33	32	ralbiia 2962	. . . . . 6 ⊢ (∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴)) ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
34	14, 33	syl6bb 275	. . . . 5 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
35	13, 34	anbi12d 743	. . . 4 ⊢ (𝐽 ∈ Top → ((𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐴) ∈ 𝐽) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))))
36	35	adantr 480	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐴) ∈ 𝐽) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))))
37		ralunb 3756	. . . 4 ⊢ (∀𝑥 ∈ (𝐴 ∪ (𝑋 ∖ 𝐴))∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
38		simpr 476	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
39		undif 4001	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∪ (𝑋 ∖ 𝐴)) = 𝑋)
40	38, 39	sylib 207	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∪ (𝑋 ∖ 𝐴)) = 𝑋)
41	40	raleqdv 3121	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∀𝑥 ∈ (𝐴 ∪ (𝑋 ∖ 𝐴))∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
42	37, 41	syl5bbr 273	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
43	4, 36, 42	3bitrd 293	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐴 ∈ 𝐽 ∧ 𝐴 ∈ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
44	1, 43	syl5bb 271	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927  df-top 20521  df-cld 20633

This theorem is referenced by:  isclo2  20702  cvmliftmolem2  30518  cvmlift2lem12  30550