Theorem connsuba 23444

Description: Connectedness for a subspace. See connsub 23445. (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 23185	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
2		dfconn2 23443	. . 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 3482	. . . . 5 ⊢ 𝑥 ∈ V
5	4	inex1 5323	. . . 4 ⊢ (𝑥 ∩ 𝐴) ∈ V
6	5	a1i 11	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ V)
7		toponmax 22948	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
8	7	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
9		simpr 484	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
10	8, 9	ssexd 5330	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
11		elrest 17474	. . . 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 3482	. . . . . 6 ⊢ 𝑦 ∈ V
14	13	inex1 5323	. . . . 5 ⊢ (𝑦 ∩ 𝐴) ∈ V
15	14	a1i 11	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑦 ∈ 𝐽) → (𝑦 ∩ 𝐴) ∈ V)
16		elrest 17474	. . . . . 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 480	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) → (𝑣 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
19		simplr 769	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → 𝑢 = (𝑥 ∩ 𝐴))
20	19	neeq1d 2998	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ≠ ∅ ↔ (𝑥 ∩ 𝐴) ≠ ∅))
21		simpr 484	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → 𝑣 = (𝑦 ∩ 𝐴))
22	21	neeq1d 2998	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑣 ≠ ∅ ↔ (𝑦 ∩ 𝐴) ≠ ∅))
23	19, 21	ineq12d 4229	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∩ 𝑣) = ((𝑥 ∩ 𝐴) ∩ (𝑦 ∩ 𝐴)))
24		inindir 4244	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ((𝑥 ∩ 𝐴) ∩ (𝑦 ∩ 𝐴))
25	23, 24	eqtr4di 2793	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∩ 𝑣) = ((𝑥 ∩ 𝑦) ∩ 𝐴))
26	25	eqeq1d 2737	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → ((𝑢 ∩ 𝑣) = ∅ ↔ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅))
27	20, 22, 26	3anbi123d 1435	. . . . 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 4179	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∪ 𝑣) = ((𝑥 ∩ 𝐴) ∪ (𝑦 ∩ 𝐴)))
29		indir 4292	. . . . . . 7 ⊢ ((𝑥 ∪ 𝑦) ∩ 𝐴) = ((𝑥 ∩ 𝐴) ∪ (𝑦 ∩ 𝐴))
30	28, 29	eqtr4di 2793	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∪ 𝑣) = ((𝑥 ∪ 𝑦) ∩ 𝐴))
31	30	neeq1d 2998	. . . . 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 5416	. . 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 5416	. 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 279	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  ‘cfv 6563  (class class class)co 7431   ↾_t crest 17467  TopOnctopon 22932  Conncconn 23435

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-conn 23436

This theorem is referenced by:  connsub  23445  nconnsubb  23447