Theorem isclo 21697

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 4171	. 2 ⊢ (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ (𝐴 ∈ 𝐽 ∧ 𝐴 ∈ (Clsd‘𝐽)))
2		isclo.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld2 21638	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝐴) ∈ 𝐽))
4	3	anbi2d 630	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐴 ∈ 𝐽 ∧ 𝐴 ∈ (Clsd‘𝐽)) ↔ (𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐴) ∈ 𝐽)))
5		eltop2 21585	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
6		dfss3 3958	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴)
7		pm5.501 369	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
8	7	ralbidv 3199	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴 ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
9	6, 8	syl5bb 285	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
10	9	anbi2d 630	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
11	10	rexbidv 3299	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
12	11	ralbiia 3166	. . . . . 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 289	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
14		eltop2 21585	. . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴))))
15		dfss3 3958	. . . . . . . . . 10 ⊢ (𝑦 ⊆ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ (𝑋 ∖ 𝐴))
16		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑦)
17		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐽)
18		elunii 4845	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐽) → 𝑧 ∈ ∪ 𝐽)
19	16, 17, 18	syl2anr 598	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ ∪ 𝐽)
20	19, 2	eleqtrrdi 2926	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑋)
21		eldif 3948	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑋 ∖ 𝐴) ↔ (𝑧 ∈ 𝑋 ∧ ¬ 𝑧 ∈ 𝐴))
22	21	baib 538	. . . . . . . . . . . . 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 4106	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
25		nbn2 373	. . . . . . . . . . . . . 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 724	. . . . . . . . . . . 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 3198	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → (∀𝑧 ∈ 𝑦 𝑧 ∈ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
30	15, 29	syl5bb 285	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → (𝑦 ⊆ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
31	30	anbi2d 630	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
32	31	rexbidva 3298	. . . . . . 7 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) → (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴)) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
33	32	ralbiia 3166	. . . . . 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 289	. . . . 5 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
35	13, 34	anbi12d 632	. . . 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 483	. . 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 4169	. . . 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 487	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
39		undif 4432	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∪ (𝑋 ∖ 𝐴)) = 𝑋)
40	38, 39	sylib 220	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∪ (𝑋 ∖ 𝐴)) = 𝑋)
41	40	raleqdv 3417	. . . 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 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))))
43	4, 36, 42	3bitrd 307	. 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 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 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  ∪ cuni 4840  ‘cfv 6357  Topctop 21503  Clsdccld 21626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-topgen 16719  df-top 21504  df-cld 21629

This theorem is referenced by:  isclo2  21698  cvmliftmolem2  32531  cvmlift2lem12  32563