Theorem isclo 20891

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 3796	. 2 ⊢ (𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ (𝐴 ∈ 𝐽 ∧ 𝐴 ∈ (Clsd‘𝐽)))
2		isclo.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld2 20832	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝐴) ∈ 𝐽))
4	3	anbi2d 740	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐴 ∈ 𝐽 ∧ 𝐴 ∈ (Clsd‘𝐽)) ↔ (𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐴) ∈ 𝐽)))
5		eltop2 20779	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))
6		dfss3 3592	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴)
7		pm5.501 356	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
8	7	ralbidv 2986	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (∀𝑧 ∈ 𝑦 𝑧 ∈ 𝐴 ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
9	6, 8	syl5bb 272	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
10	9	anbi2d 740	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
11	10	rexbidv 3052	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
12	11	ralbiia 2979	. . . . . 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 276	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
14		eltop2 20779	. . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴))))
15		dfss3 3592	. . . . . . . . . 10 ⊢ (𝑦 ⊆ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 𝑧 ∈ (𝑋 ∖ 𝐴))
16		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑦)
17		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐽)
18		elunii 4441	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐽) → 𝑧 ∈ ∪ 𝐽)
19	16, 17, 18	syl2anr 495	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ ∪ 𝐽)
20	19, 2	syl6eleqr 2712	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑋)
21		eldif 3584	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑋 ∖ 𝐴) ↔ (𝑧 ∈ 𝑋 ∧ ¬ 𝑧 ∈ 𝐴))
22	21	baib 944	. . . . . . . . . . . . 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 3733	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
25		nbn2 360	. . . . . . . . . . . . . 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 762	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → (¬ 𝑧 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
28	23, 27	bitrd 268	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) ∧ 𝑧 ∈ 𝑦) → (𝑧 ∈ (𝑋 ∖ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
29	28	ralbidva 2985	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → (∀𝑧 ∈ 𝑦 𝑧 ∈ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
30	15, 29	syl5bb 272	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → (𝑦 ⊆ (𝑋 ∖ 𝐴) ↔ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)))
31	30	anbi2d 740	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝐴) ∧ 𝑦 ∈ 𝐽) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
32	31	rexbidva 3049	. . . . . . 7 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) → (∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ (𝑋 ∖ 𝐴)) ↔ ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
33	32	ralbiia 2979	. . . . . 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 276	. . . . 5 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))))
35	13, 34	anbi12d 747	. . . 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 481	. . 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 3794	. . . 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 477	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
39		undif 4049	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∪ (𝑋 ∖ 𝐴)) = 𝑋)
40	38, 39	sylib 208	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∪ (𝑋 ∖ 𝐴)) = 𝑋)
41	40	raleqdv 3144	. . . 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 274	. . 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 294	. 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 272	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∪ cuni 4436  ‘cfv 5888  Topctop 20698  Clsdccld 20820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-topgen 16104  df-top 20699  df-cld 20823

This theorem is referenced by:  isclo2  20892  cvmliftmolem2  31264  cvmlift2lem12  31296