Theorem connsubclo 22035

Description: If a clopen set meets a connected subspace, it must contain the entire subspace. (Contributed by Mario Carneiro, 10-Mar-2015.)

Hypotheses

Ref	Expression
connsubclo.1	⊢ 𝑋 = ∪ 𝐽
connsubclo.3	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
connsubclo.4	⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ Conn)
connsubclo.5	⊢ (𝜑 → 𝐵 ∈ 𝐽)
connsubclo.6	⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅)
connsubclo.7	⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽))

Assertion

Ref	Expression
connsubclo	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Proof of Theorem connsubclo

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2824	. . . 4 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴)
2		connsubclo.4	. . . 4 ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ Conn)
3		connsubclo.7	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽))
4		cldrcl 21637	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
6		connsubclo.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
7	6	topopn 21517	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
8	5, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
9		connsubclo.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
10	8, 9	ssexd 5231	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
11		connsubclo.5	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐽)
12		elrestr 16705	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝐵 ∈ 𝐽) → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
13	5, 10, 11, 12	syl3anc 1367	. . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
14		connsubclo.6	. . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅)
15		eqid 2824	. . . . . 6 ⊢ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴)
16		ineq1 4184	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ∩ 𝐴) = (𝐵 ∩ 𝐴))
17	16	rspceeqv 3641	. . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴)) → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴))
18	3, 15, 17	sylancl 588	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴))
19	6	restcld 21783	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)))
20	5, 9, 19	syl2anc 586	. . . . 5 ⊢ (𝜑 → ((𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)))
21	18, 20	mpbird 259	. . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
22	1, 2, 13, 14, 21	connclo 22026	. . 3 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = ∪ (𝐽 ↾t 𝐴))
23	6	restuni 21773	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
24	5, 9, 23	syl2anc 586	. . 3 ⊢ (𝜑 → 𝐴 = ∪ (𝐽 ↾t 𝐴))
25	22, 24	eqtr4d 2862	. 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴)
26		sseqin2 4195	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
27	25, 26	sylibr 236	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ∃wrex 3142  Vcvv 3497   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294  ∪ cuni 4841  ‘cfv 6358  (class class class)co 7159   ↾_t crest 16697  Topctop 21504  Clsdccld 21627  Conncconn 22022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-oadd 8109  df-er 8292  df-en 8513  df-fin 8516  df-fi 8878  df-rest 16699  df-topgen 16720  df-top 21505  df-topon 21522  df-bases 21557  df-cld 21630  df-conn 22023

This theorem is referenced by:  conncn  22037  conncompclo  22046