Theorem isclo 23127

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 3920	. 2 ⊢ (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ (𝐴 ∈ 𝐽 ∧ 𝐴 ∈ (Clsd‘𝐽)))
2		isclo.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld2 23068	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝐴) ∈ 𝐽))
4	3	anbi2d 639	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐴 ∈ 𝐽 ∧ 𝐴 ∈ (Clsd‘𝐽)) ↔ (𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐴) ∈ 𝐽)))
5		eltop2 23015	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
6		dfss3 3925	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴)
7		pm5.501 368	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
8	7	ralbidv 3184	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴 ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
9	6, 8	bitrid 285	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
10	9	anbi2d 639	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
11	10	rexbidv 3185	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
12	11	ralbiia 3105	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
13	5, 12	bitrdi 289	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
14		eltop2 23015	. . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴))))
15		dfss3 3925	. . . . . . . . . 10 ⊢ (𝑦 ⊆ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ (𝑋 ∖ 𝐴))
16		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑦)
17		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐽)
18		elunii 4869	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐽) → 𝑧 ∈ ∪ 𝐽)
19	16, 17, 18	syl2anr 606	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ ∪ 𝐽)
20	19, 2	eleqtrrdi 2872	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑋)
21		eldif 3914	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑋 ∖ 𝐴) ↔ (𝑧 ∈ 𝑋 ∧ ¬ 𝑧 ∈ 𝐴))
22	21	baib 543	. . . . . . . . . . . . 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 4085	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
25		nbn2 372	. . . . . . . . . . . . . 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 736	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → (¬ 𝑧 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
28	23, 27	bitrd 281	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → (𝑧 ∈ (𝑋 ∖ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
29	28	ralbidva 3182	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → (∀𝑧 ∈ 𝑦 𝑧 ∈ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
30	15, 29	bitrid 285	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → (𝑦 ⊆ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
31	30	anbi2d 639	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
32	31	rexbidva 3183	. . . . . . 7 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) → (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴)) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
33	32	ralbiia 3105	. . . . . 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	bitrdi 289	. . . . 5 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
35	13, 34	anbi12d 641	. . . 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 484	. . 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 4149	. . . 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		undif 4435	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∪ (𝑋 ∖ 𝐴)) = 𝑋)
39	38	bilani 508	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∪ (𝑋 ∖ 𝐴)) = 𝑋)
40	39	raleqdv 3319	. . . 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))))
41	37, 40	bitr3id 287	. . 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))))
42	4, 36, 41	3bitrd 307	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐴 ∈ 𝐽 ∧ 𝐴 ∈ (Clsd‘𝐽)) ↔ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
43	1, 42	bitrid 285	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 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∪ cuni 4864  ‘cfv 6517  Topctop 22933  Clsdccld 23056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-topgen 17455  df-top 22934  df-cld 23059

This theorem is referenced by:  isclo2  23128  cvmliftmolem2  35596  cvmlift2lem12  35628