Theorem connsuba 23685

Description: Connectedness for a subspace. See connsub 23686. (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 23426	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
2		dfconn2 23684	. . 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 18	. 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 3454	. . . . 5 ⊢ 𝑥 ∈ V
5	4	inex1 5277	. . . 4 ⊢ (𝑥 ∩ 𝐴) ∈ V
6	5	a1i 11	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ V)
7		toponmax 23191	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
8	7	adantr 486	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
9		simpr 490	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
10	8, 9	ssexd 5286	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
11		elrest 17545	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑢 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑢 = (𝑥 ∩ 𝐴)))
12	10, 11	syldan 603	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑢 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑢 = (𝑥 ∩ 𝐴)))
13		vex 3454	. . . . . 6 ⊢ 𝑦 ∈ V
14	13	inex1 5277	. . . . 5 ⊢ (𝑦 ∩ 𝐴) ∈ V
15	14	a1i 11	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑦 ∈ 𝐽) → (𝑦 ∩ 𝐴) ∈ V)
16		elrest 17545	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑣 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
17	10, 16	syldan 603	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑣 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
18	17	adantr 486	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) → (𝑣 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
19		simplr 781	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → 𝑢 = (𝑥 ∩ 𝐴))
20	19	neeq1d 3014	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ≠ ∅ ↔ (𝑥 ∩ 𝐴) ≠ ∅))
21		simpr 490	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → 𝑣 = (𝑦 ∩ 𝐴))
22	21	neeq1d 3014	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑣 ≠ ∅ ↔ (𝑦 ∩ 𝐴) ≠ ∅))
23	19, 21	ineq12d 4167	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∩ 𝑣) = ((𝑥 ∩ 𝐴) ∩ (𝑦 ∩ 𝐴)))
24		inindir 4181	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ((𝑥 ∩ 𝐴) ∩ (𝑦 ∩ 𝐴))
25	23, 24	eqtr4di 2813	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∩ 𝑣) = ((𝑥 ∩ 𝑦) ∩ 𝐴))
26	25	eqeq1d 2762	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → ((𝑢 ∩ 𝑣) = ∅ ↔ ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅))
27	20, 22, 26	3anbi123d 1464	. . . . 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 4116	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∪ 𝑣) = ((𝑥 ∩ 𝐴) ∪ (𝑦 ∩ 𝐴)))
29		indir 4232	. . . . . . 7 ⊢ ((𝑥 ∪ 𝑦) ∩ 𝐴) = ((𝑥 ∩ 𝐴) ∪ (𝑦 ∩ 𝐴))
30	28, 29	eqtr4di 2813	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∪ 𝑣) = ((𝑥 ∪ 𝑦) ∩ 𝐴))
31	30	neeq1d 3014	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → ((𝑢 ∪ 𝑣) ≠ 𝐴 ↔ ((𝑥 ∪ 𝑦) ∩ 𝐴) ≠ 𝐴))
32	27, 31	imbi12d 347	. . . 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 5372	. . 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 5372	. 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 282	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ‘cfv 6528  (class class class)co 7409   ↾_t crest 17538  TopOnctopon 23175  Conncconn 23676

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7862  df-1st 7985  df-2nd 7986  df-en 8953  df-fin 8956  df-fi 9381  df-rest 17540  df-topgen 17561  df-top 23159  df-topon 23176  df-bases 23211  df-cld 23284  df-conn 23677

This theorem is used by:  connsub  23686  nconnsubb  23688