Theorem isclo 21694

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 4168	. 2 ⊢ (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ (𝐴 ∈ 𝐽 ∧ 𝐴 ∈ (Clsd‘𝐽)))
2		isclo.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld2 21635	. . . 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 21582	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
6		dfss3 3955	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴)
7		pm5.501 369	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
8	7	ralbidv 3197	. . . . . . . . . 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 3297	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
12	11	ralbiia 3164	. . . . . 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 21582	. . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴))))
15		dfss3 3955	. . . . . . . . . 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 4842	. . . . . . . . . . . . . . 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 2924	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑋)
21		eldif 3945	. . . . . . . . . . . . . 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 4103	. . . . . . . . . . . . . 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 3196	. . . . . . . . . 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 3296	. . . . . . 7 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) → (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴)) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
33	32	ralbiia 3164	. . . . . 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 4166	. . . 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 4429	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∪ (𝑋 ∖ 𝐴)) = 𝑋)
40	38, 39	sylib 220	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∪ (𝑋 ∖ 𝐴)) = 𝑋)
41	40	raleqdv 3415	. . . 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 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ∖ cdif 3932   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935  ∪ cuni 4837  ‘cfv 6354  Topctop 21500  Clsdccld 21623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-topgen 16716  df-top 21501  df-cld 21626

This theorem is referenced by:  isclo2  21695  cvmliftmolem2  32529  cvmlift2lem12  32561