Theorem tgrest 14405

Description: A subspace can be generated by restricted sets from a basis for the original topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
tgrest	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (topGen‘(𝐵 ↾t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))

Proof of Theorem tgrest

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		restfn 12914	. . . . . 6 ⊢ ↾t Fn (V × V)
2		elex 2774	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
3		elex 2774	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
4		fnovex 5955	. . . . . 6 ⊢ (( ↾t Fn (V × V) ∧ 𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) ∈ V)
5	1, 2, 3, 4	mp3an3an 1354	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐵 ↾t 𝐴) ∈ V)
6		eltg3 14293	. . . . 5 ⊢ ((𝐵 ↾t 𝐴) ∈ V → (𝑥 ∈ (topGen‘(𝐵 ↾t 𝐴)) ↔ ∃𝑦(𝑦 ⊆ (𝐵 ↾t 𝐴) ∧ 𝑥 = ∪ 𝑦)))
7	5, 6	syl 14	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (topGen‘(𝐵 ↾t 𝐴)) ↔ ∃𝑦(𝑦 ⊆ (𝐵 ↾t 𝐴) ∧ 𝑥 = ∪ 𝑦)))
8		simpll 527	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → 𝐵 ∈ 𝑉)
9		funmpt 5296	. . . . . . . . . 10 ⊢ Fun (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴))
10	9	a1i 9	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → Fun (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)))
11		restval 12916	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐵 ↾t 𝐴) = ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)))
12	11	sseq2d 3213	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑦 ⊆ (𝐵 ↾t 𝐴) ↔ 𝑦 ⊆ ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴))))
13	12	biimpa 296	. . . . . . . . . 10 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → 𝑦 ⊆ ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)))
14		vex 2766	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
15	14	inex1 4167	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝐴) ∈ V
16	15	rgenw 2552	. . . . . . . . . . 11 ⊢ ∀𝑥 ∈ 𝐵 (𝑥 ∩ 𝐴) ∈ V
17		eqid 2196	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) = (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴))
18	17	fnmpt 5384	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∩ 𝐴) ∈ V → (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) Fn 𝐵)
19		fnima 5376	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) Fn 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝐵) = ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)))
20	16, 18, 19	mp2b 8	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝐵) = ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴))
21	13, 20	sseqtrrdi 3232	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → 𝑦 ⊆ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝐵))
22		ssimaexg 5623	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ Fun (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ∧ 𝑦 ⊆ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝐵)) → ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)))
23	8, 10, 21, 22	syl3anc 1249	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)))
24		df-ima 4676	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ran ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧)
25		resmpt 4994	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧) = (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
26	25	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧) = (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
27	26	rneqd 4895	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ran ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧) = ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
28	24, 27	eqtrid 2241	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
29	28	unieqd 3850	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ∪ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
30	15	dfiun3 4925	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴) = ∪ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴))
31	29, 30	eqtr4di 2247	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴))
32		iunin1 3981	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴) = (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴)
33	31, 32	eqtrdi 2245	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴))
34		tgvalex 12934	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V)
35	34	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (topGen‘𝐵) ∈ V)
36		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝐴 ∈ 𝑊)
37	36	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → 𝐴 ∈ 𝑊)
38		uniiun 3970	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 = ∪ 𝑥 ∈ 𝑧 𝑥
39		eltg3i 14292	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑧 ∈ (topGen‘𝐵))
40	38, 39	eqeltrrid 2284	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑥 ∈ 𝑧 𝑥 ∈ (topGen‘𝐵))
41	40	adantlr 477	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑥 ∈ 𝑧 𝑥 ∈ (topGen‘𝐵))
42		elrestr 12918	. . . . . . . . . . . . . 14 ⊢ (((topGen‘𝐵) ∈ V ∧ 𝐴 ∈ 𝑊 ∧ ∪ 𝑥 ∈ 𝑧 𝑥 ∈ (topGen‘𝐵)) → (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
43	35, 37, 41, 42	syl3anc 1249	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
44	33, 43	eqeltrd 2273	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴))
45		unieq 3848	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) → ∪ 𝑦 = ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧))
46	45	eleq1d 2265	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) → (∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴)))
47	44, 46	syl5ibrcom 157	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
48	47	expimpd 363	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
49	48	exlimdv 1833	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
50	49	adantr 276	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → (∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
51	23, 50	mpd 13	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴))
52		eleq1 2259	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
53	51, 52	syl5ibrcom 157	. . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → (𝑥 = ∪ 𝑦 → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
54	53	expimpd 363	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑦 ⊆ (𝐵 ↾t 𝐴) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
55	54	exlimdv 1833	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑦(𝑦 ⊆ (𝐵 ↾t 𝐴) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
56	7, 55	sylbid 150	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (topGen‘(𝐵 ↾t 𝐴)) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
57	56	ssrdv 3189	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (topGen‘(𝐵 ↾t 𝐴)) ⊆ ((topGen‘𝐵) ↾t 𝐴))
58		restval 12916	. . . 4 ⊢ (((topGen‘𝐵) ∈ V ∧ 𝐴 ∈ 𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)))
59	34, 36, 58	syl2an2r 595	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)))
60		eltg3 14293	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧)))
61	60	adantr 276	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧)))
62	38	ineq1i 3360	. . . . . . . . . . . 12 ⊢ (∪ 𝑧 ∩ 𝐴) = (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴)
63	62, 32	eqtr4i 2220	. . . . . . . . . . 11 ⊢ (∪ 𝑧 ∩ 𝐴) = ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴)
64		simplll 533	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → 𝐵 ∈ 𝑉)
65		simpllr 534	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → 𝐴 ∈ 𝑊)
66		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → 𝑧 ⊆ 𝐵)
67	66	sselda 3183	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝐵)
68		elrestr 12918	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
69	64, 65, 67, 68	syl3anc 1249	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → (𝑥 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
70	69	fmpttd 5717	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)):𝑧⟶(𝐵 ↾t 𝐴))
71	70	frnd 5417	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)) ⊆ (𝐵 ↾t 𝐴))
72		eltg3i 14292	. . . . . . . . . . . . 13 ⊢ (((𝐵 ↾t 𝐴) ∈ V ∧ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)) ⊆ (𝐵 ↾t 𝐴)) → ∪ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)) ∈ (topGen‘(𝐵 ↾t 𝐴)))
73	5, 71, 72	syl2an2r 595	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)) ∈ (topGen‘(𝐵 ↾t 𝐴)))
74	30, 73	eqeltrid 2283	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)))
75	63, 74	eqeltrid 2283	. . . . . . . . . 10 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (∪ 𝑧 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)))
76		ineq1 3357	. . . . . . . . . . 11 ⊢ (𝑤 = ∪ 𝑧 → (𝑤 ∩ 𝐴) = (∪ 𝑧 ∩ 𝐴))
77	76	eleq1d 2265	. . . . . . . . . 10 ⊢ (𝑤 = ∪ 𝑧 → ((𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)) ↔ (∪ 𝑧 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
78	75, 77	syl5ibrcom 157	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (𝑤 = ∪ 𝑧 → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
79	78	expimpd 363	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
80	79	exlimdv 1833	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
81	61, 80	sylbid 150	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑤 ∈ (topGen‘𝐵) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
82	81	imp 124	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑤 ∈ (topGen‘𝐵)) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)))
83	82	fmpttd 5717	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)):(topGen‘𝐵)⟶(topGen‘(𝐵 ↾t 𝐴)))
84	83	frnd 5417	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)) ⊆ (topGen‘(𝐵 ↾t 𝐴)))
85	59, 84	eqsstrd 3219	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((topGen‘𝐵) ↾t 𝐴) ⊆ (topGen‘(𝐵 ↾t 𝐴)))
86	57, 85	eqssd 3200	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (topGen‘(𝐵 ↾t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ∩ cin 3156   ⊆ wss 3157  ∪ cuni 3839  ∪ ciun 3916   ↦ cmpt 4094   × cxp 4661  ran crn 4664   ↾ cres 4665   “ cima 4666  Fun wfun 5252   Fn wfn 5253  ‘cfv 5258  (class class class)co 5922   ↾_t crest 12910  topGenctg 12925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-rest 12912  df-topgen 12931

This theorem is referenced by:  resttop  14406  txrest  14512