Theorem connsubclo 22483

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 2738	. . . 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 22085	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
6		connsubclo.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
7	6	topopn 21963	. . . . . . 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 5243	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
11		connsubclo.5	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐽)
12		elrestr 17056	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝐵 ∈ 𝐽) → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
13	5, 10, 11, 12	syl3anc 1369	. . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
14		connsubclo.6	. . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅)
15		eqid 2738	. . . . . 6 ⊢ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴)
16		ineq1 4136	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ∩ 𝐴) = (𝐵 ∩ 𝐴))
17	16	rspceeqv 3567	. . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝐵 ∩ 𝐴) = (𝐵 ∩ 𝐴)) → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴))
18	3, 15, 17	sylancl 585	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴))
19	6	restcld 22231	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)))
20	5, 9, 19	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵 ∩ 𝐴) = (𝑥 ∩ 𝐴)))
21	18, 20	mpbird 256	. . . 4 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
22	1, 2, 13, 14, 21	connclo 22474	. . 3 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = ∪ (𝐽 ↾t 𝐴))
23	6	restuni 22221	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
24	5, 9, 23	syl2anc 583	. . 3 ⊢ (𝜑 → 𝐴 = ∪ (𝐽 ↾t 𝐴))
25	22, 24	eqtr4d 2781	. 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴)
26		sseqin2 4146	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
27	25, 26	sylibr 233	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  ‘cfv 6418  (class class class)co 7255   ↾_t crest 17048  Topctop 21950  Clsdccld 22075  Conncconn 22470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-en 8692  df-fin 8695  df-fi 9100  df-rest 17050  df-topgen 17071  df-top 21951  df-topon 21968  df-bases 22004  df-cld 22078  df-conn 22471

This theorem is referenced by:  conncn  22485  conncompclo  22494