Theorem connsuba 22771

Description: Connectedness for a subspace. See connsub 22772. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)

Assertion

Ref	Expression
connsuba	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐽 ↾t 𝐴) ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦

Proof of Theorem connsuba

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resttopon 22512	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
2		dfconn2 22770	. . 3 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) → ((𝐽 ↾t 𝐴) ∈ Conn ↔ ∀𝑢 ∈ (𝐽 ↾t 𝐴)∀𝑣 ∈ (𝐽 ↾t 𝐴)((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑢 ∪ 𝑣) ≠ 𝐴)))
3	1, 2	syl 17	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐽 ↾t 𝐴) ∈ Conn ↔ ∀𝑢 ∈ (𝐽 ↾t 𝐴)∀𝑣 ∈ (𝐽 ↾t 𝐴)((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑢 ∪ 𝑣) ≠ 𝐴)))
4		vex 3449	. . . . 5 ⊢ 𝑥 ∈ V
5	4	inex1 5274	. . . 4 ⊢ (𝑥 ∩ 𝐴) ∈ V
6	5	a1i 11	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ V)
7		toponmax 22275	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
8	7	adantr 481	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
9		simpr 485	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
10	8, 9	ssexd 5281	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
11		elrest 17309	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑢 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑢 = (𝑥 ∩ 𝐴)))
12	10, 11	syldan 591	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑢 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑢 = (𝑥 ∩ 𝐴)))
13		vex 3449	. . . . . 6 ⊢ 𝑦 ∈ V
14	13	inex1 5274	. . . . 5 ⊢ (𝑦 ∩ 𝐴) ∈ V
15	14	a1i 11	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑦 ∈ 𝐽) → (𝑦 ∩ 𝐴) ∈ V)
16		elrest 17309	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑣 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
17	10, 16	syldan 591	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑣 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
18	17	adantr 481	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) → (𝑣 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
19		simplr 767	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → 𝑢 = (𝑥 ∩ 𝐴))
20	19	neeq1d 3003	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ≠ ∅ ↔ (𝑥 ∩ 𝐴) ≠ ∅))
21		simpr 485	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → 𝑣 = (𝑦 ∩ 𝐴))
22	21	neeq1d 3003	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑣 ≠ ∅ ↔ (𝑦 ∩ 𝐴) ≠ ∅))
23	19, 21	ineq12d 4173	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∩ 𝑣) = ((𝑥 ∩ 𝐴) ∩ (𝑦 ∩ 𝐴)))
24		inindir 4187	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ((𝑥 ∩ 𝐴) ∩ (𝑦 ∩ 𝐴))
25	23, 24	eqtr4di 2794	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∩ 𝑣) = ((𝑥 ∩ 𝑦) ∩ 𝐴))
26	25	eqeq1d 2738	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → ((𝑢 ∩ 𝑣) = ∅ ↔ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅))
27	20, 22, 26	3anbi123d 1436	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → ((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢 ∩ 𝑣) = ∅) ↔ ((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅)))
28	19, 21	uneq12d 4124	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∪ 𝑣) = ((𝑥 ∩ 𝐴) ∪ (𝑦 ∩ 𝐴)))
29		indir 4235	. . . . . . 7 ⊢ ((𝑥 ∪ 𝑦) ∩ 𝐴) = ((𝑥 ∩ 𝐴) ∪ (𝑦 ∩ 𝐴))
30	28, 29	eqtr4di 2794	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∪ 𝑣) = ((𝑥 ∪ 𝑦) ∩ 𝐴))
31	30	neeq1d 3003	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → ((𝑢 ∪ 𝑣) ≠ 𝐴 ↔ ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴))
32	27, 31	imbi12d 344	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑢 ∪ 𝑣) ≠ 𝐴) ↔ (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴)))
33	15, 18, 32	ralxfr2d 5365	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) → (∀𝑣 ∈ (𝐽 ↾t 𝐴)((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑢 ∪ 𝑣) ≠ 𝐴) ↔ ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴)))
34	6, 12, 33	ralxfr2d 5365	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (∀𝑢 ∈ (𝐽 ↾t 𝐴)∀𝑣 ∈ (𝐽 ↾t 𝐴)((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑢 ∪ 𝑣) ≠ 𝐴) ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴)))
35	3, 34	bitrd 278	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐽 ↾t 𝐴) ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝐴) ≠ ∅ ∧ (𝑦 ∩ 𝐴) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  ‘cfv 6496  (class class class)co 7357   ↾_t crest 17302  TopOnctopon 22259  Conncconn 22762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-en 8884  df-fin 8887  df-fi 9347  df-rest 17304  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-cld 22370  df-conn 22763

This theorem is referenced by:  connsub  22772  nconnsubb  22774