Theorem connsuba 23368

Description: Connectedness for a subspace. See connsub 23369. (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 23109	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
2		dfconn2 23367	. . 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 3445	. . . . 5 ⊢ 𝑥 ∈ V
5	4	inex1 5263	. . . 4 ⊢ (𝑥 ∩ 𝐴) ∈ V
6	5	a1i 11	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ V)
7		toponmax 22874	. . . . . 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 5270	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
11		elrest 17351	. . . 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 3445	. . . . . 6 ⊢ 𝑦 ∈ V
14	13	inex1 5263	. . . . 5 ⊢ (𝑦 ∩ 𝐴) ∈ V
15	14	a1i 11	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑦 ∈ 𝐽) → (𝑦 ∩ 𝐴) ∈ V)
16		elrest 17351	. . . . . 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 4174	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∩ 𝑣) = ((𝑥 ∩ 𝐴) ∩ (𝑦 ∩ 𝐴)))
24		inindir 4189	. . . . . . . 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 4122	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑢 = (𝑥 ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) → (𝑢 ∪ 𝑣) = ((𝑥 ∩ 𝐴) ∪ (𝑦 ∩ 𝐴)))
29		indir 4239	. . . . . . 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 5356	. . 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 5356	. 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 3061  Vcvv 3441   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286  ‘cfv 6493  (class class class)co 7360   ↾_t crest 17344  TopOnctopon 22858  Conncconn 23359

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-en 8888  df-fin 8891  df-fi 9318  df-rest 17346  df-topgen 17367  df-top 22842  df-topon 22859  df-bases 22894  df-cld 22967  df-conn 23360

This theorem is referenced by:  connsub  23369  nconnsubb  23371