Theorem tgrest 23510

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		ovex 7479	. . . . 5 ⊢ (𝐵 ↾t 𝐴) ∈ V
2		eltg3 23313	. . . . 5 ⊢ ((𝐵 ↾t 𝐴) ∈ V → (𝑥 ∈ (topGen‘(𝐵 ↾t 𝐴)) ↔ ∃𝑦(𝑦 ⊆ (𝐵 ↾t 𝐴) ∧ 𝑥 = ∪ 𝑦)))
3	1, 2	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ (topGen‘(𝐵 ↾t 𝐴)) ↔ ∃𝑦(𝑦 ⊆ (𝐵 ↾t 𝐴) ∧ 𝑥 = ∪ 𝑦))
4		simpll 779	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → 𝐵 ∈ 𝑉)
5		funmpt 6601	. . . . . . . . . 10 ⊢ Fun (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴))
6	5	a1i 11	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → Fun (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)))
7		restval 17629	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐵 ↾t 𝐴) = ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)))
8	7	sseq2d 3945	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑦 ⊆ (𝐵 ↾t 𝐴) ↔ 𝑦 ⊆ ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴))))
9	8	biimpa 482	. . . . . . . . . 10 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → 𝑦 ⊆ ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)))
10		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
11	10	inex1 5282	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝐴) ∈ V
12	11	rgenw 3084	. . . . . . . . . . 11 ⊢ ∀𝑥 ∈ 𝐵 (𝑥 ∩ 𝐴) ∈ V
13		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) = (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴))
14	13	fnmpt 6704	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∩ 𝐴) ∈ V → (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) Fn 𝐵)
15		fnima 6694	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) Fn 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝐵) = ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)))
16	12, 14, 15	mp2b 10	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝐵) = ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴))
17	9, 16	sseqtrrdi 3954	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → 𝑦 ⊆ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝐵))
18		ssimaexg 6997	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ Fun (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ∧ 𝑦 ⊆ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝐵)) → ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)))
19	4, 6, 17, 18	syl3anc 1399	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)))
20		df-ima 5669	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ran ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧)
21		resmpt 6042	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧) = (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
22	21	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧) = (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
23	22	rneqd 5928	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ran ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧) = ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
24	20, 23	eqtrid 2810	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
25	24	unieqd 4883	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ∪ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
26	11	dfiun3 5960	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴) = ∪ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴))
27	25, 26	eqtr4di 2816	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴))
28		iunin1 5033	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴) = (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴)
29	27, 28	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴))
30		fvex 6924	. . . . . . . . . . . . . 14 ⊢ (topGen‘𝐵) ∈ V
31		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝐴 ∈ 𝑊)
32		uniiun 5020	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 = ∪ 𝑥 ∈ 𝑧 𝑥
33		eltg3i 23312	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑧 ∈ (topGen‘𝐵))
34	32, 33	eqeltrrid 2869	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑥 ∈ 𝑧 𝑥 ∈ (topGen‘𝐵))
35	34	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑥 ∈ 𝑧 𝑥 ∈ (topGen‘𝐵))
36		elrestr 17631	. . . . . . . . . . . . . 14 ⊢ (((topGen‘𝐵) ∈ V ∧ 𝐴 ∈ 𝑊 ∧ ∪ 𝑥 ∈ 𝑧 𝑥 ∈ (topGen‘𝐵)) → (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
37	30, 31, 35, 36	mp3an2ani 1498	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
38	29, 37	eqeltrd 2864	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴))
39		unieq 4881	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) → ∪ 𝑦 = ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧))
40	39	eleq1d 2849	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) → (∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴)))
41	38, 40	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
42	41	expimpd 459	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
43	42	exlimdv 1968	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
44	43	adantr 486	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → (∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
45	19, 44	mpd 16	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴))
46		eleq1 2852	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
47	45, 46	syl5ibrcom 250	. . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → (𝑥 = ∪ 𝑦 → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
48	47	expimpd 459	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑦 ⊆ (𝐵 ↾t 𝐴) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
49	48	exlimdv 1968	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑦(𝑦 ⊆ (𝐵 ↾t 𝐴) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
50	3, 49	biimtrid 245	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (topGen‘(𝐵 ↾t 𝐴)) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
51	50	ssrdv 3919	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (topGen‘(𝐵 ↾t 𝐴)) ⊆ ((topGen‘𝐵) ↾t 𝐴))
52		restval 17629	. . . 4 ⊢ (((topGen‘𝐵) ∈ V ∧ 𝐴 ∈ 𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)))
53	30, 31, 52	sylancr 599	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)))
54		eltg3 23313	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧)))
55	54	adantr 486	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧)))
56	32	ineq1i 4116	. . . . . . . . . . . 12 ⊢ (∪ 𝑧 ∩ 𝐴) = (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴)
57	56, 28	eqtr4i 2789	. . . . . . . . . . 11 ⊢ (∪ 𝑧 ∩ 𝐴) = ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴)
58		simplll 787	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → 𝐵 ∈ 𝑉)
59		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → 𝐴 ∈ 𝑊)
60		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → 𝑧 ⊆ 𝐵)
61	60	sselda 3913	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝐵)
62		elrestr 17631	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
63	58, 59, 61, 62	syl3anc 1399	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → (𝑥 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
64	63	fmpttd 7142	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)):𝑧⟶(𝐵 ↾t 𝐴))
65	64	frnd 6743	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)) ⊆ (𝐵 ↾t 𝐴))
66		eltg3i 23312	. . . . . . . . . . . . 13 ⊢ (((𝐵 ↾t 𝐴) ∈ V ∧ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)) ⊆ (𝐵 ↾t 𝐴)) → ∪ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)) ∈ (topGen‘(𝐵 ↾t 𝐴)))
67	1, 65, 66	sylancr 599	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)) ∈ (topGen‘(𝐵 ↾t 𝐴)))
68	26, 67	eqeltrid 2868	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)))
69	57, 68	eqeltrid 2868	. . . . . . . . . 10 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (∪ 𝑧 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)))
70		ineq1 4113	. . . . . . . . . . 11 ⊢ (𝑤 = ∪ 𝑧 → (𝑤 ∩ 𝐴) = (∪ 𝑧 ∩ 𝐴))
71	70	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑤 = ∪ 𝑧 → ((𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)) ↔ (∪ 𝑧 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
72	69, 71	syl5ibrcom 250	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (𝑤 = ∪ 𝑧 → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
73	72	expimpd 459	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
74	73	exlimdv 1968	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
75	55, 74	sylbid 243	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑤 ∈ (topGen‘𝐵) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
76	75	imp 412	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑤 ∈ (topGen‘𝐵)) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)))
77	76	fmpttd 7142	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)):(topGen‘𝐵)⟶(topGen‘(𝐵 ↾t 𝐴)))
78	77	frnd 6743	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)) ⊆ (topGen‘(𝐵 ↾t 𝐴)))
79	53, 78	eqsstrd 3947	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((topGen‘𝐵) ↾t 𝐴) ⊆ (topGen‘(𝐵 ↾t 𝐴)))
80	51, 79	eqssd 3930	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (topGen‘(𝐵 ↾t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5190  ran crn 5657   ↾ cres 5658   “ cima 5659  Fun wfun 6555   Fn wfn 6556  ‘cfv 6561  (class class class)co 7446   ↾_t crest 17623  topGenctg 17640

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-rest 17625  df-topgen 17646

This theorem is used by:  resttop  23511  ordtrest2  23555  2ndcrest  23805  txrest  23983  xkoptsub  24006  xrtgioo  25159  ordtrest2NEW  34638  ptrest  38742