Theorem connsuba 23399

Description: Connectedness for a subspace. See connsub 23400. (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 23140	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
2		dfconn2 23398	. . 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 3434	. . . . 5 ⊢ 𝑥 ∈ V
5	4	inex1 5255	. . . 4 ⊢ (𝑥 ∩ 𝐴) ∈ V
6	5	a1i 11	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ V)
7		toponmax 22905	. . . . . 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 5262	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
11		elrest 17385	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑢 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑢 = (𝑥 ∩ 𝐴)))
12	10, 11	syldan 592	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑢 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑢 = (𝑥 ∩ 𝐴)))
13		vex 3434	. . . . . 6 ⊢ 𝑦 ∈ V
14	13	inex1 5255	. . . . 5 ⊢ (𝑦 ∩ 𝐴) ∈ V
15	14	a1i 11	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑦 ∈ 𝐽) → (𝑦 ∩ 𝐴) ∈ V)
16		elrest 17385	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑣 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
17	10, 16	syldan 592	. . . . 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 2992	. . . . . 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 2992	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑣 ≠ ∅ ↔ (𝑦 ∩ 𝐴) ≠ ∅))
23	19, 21	ineq12d 4162	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∩ 𝑣) = ((𝑥 ∩ 𝐴) ∩ (𝑦 ∩ 𝐴)))
24		inindir 4177	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ((𝑥 ∩ 𝐴) ∩ (𝑦 ∩ 𝐴))
25	23, 24	eqtr4di 2790	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∩ 𝑣) = ((𝑥 ∩ 𝑦) ∩ 𝐴))
26	25	eqeq1d 2739	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → ((𝑢 ∩ 𝑣) = ∅ ↔ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅))
27	20, 22, 26	3anbi123d 1439	. . . . 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 4110	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∪ 𝑣) = ((𝑥 ∩ 𝐴) ∪ (𝑦 ∩ 𝐴)))
29		indir 4227	. . . . . . 7 ⊢ ((𝑥 ∪ 𝑦) ∩ 𝐴) = ((𝑥 ∩ 𝐴) ∪ (𝑦 ∩ 𝐴))
30	28, 29	eqtr4di 2790	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∪ 𝑣) = ((𝑥 ∪ 𝑦) ∩ 𝐴))
31	30	neeq1d 2992	. . . . 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 5349	. . 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 5349	. 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 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  ‘cfv 6494  (class class class)co 7362   ↾_t crest 17378  TopOnctopon 22889  Conncconn 23390

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-en 8889  df-fin 8892  df-fi 9319  df-rest 17380  df-topgen 17401  df-top 22873  df-topon 22890  df-bases 22925  df-cld 22998  df-conn 23391

This theorem is referenced by:  connsub  23400  nconnsubb  23402