Theorem tgrest 12120

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 11906	. . . . . 6 ⊢ ↾t Fn (V × V)
2		elex 2652	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
3		elex 2652	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
4		fnovex 5736	. . . . . 6 ⊢ (( ↾t Fn (V × V) ∧ 𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 ↾t 𝐴) ∈ V)
5	1, 2, 3, 4	mp3an3an 1289	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐵 ↾t 𝐴) ∈ V)
6		eltg3 12008	. . . . 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 499	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → 𝐵 ∈ 𝑉)
9		funmpt 5097	. . . . . . . . . 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 11908	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐵 ↾t 𝐴) = ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)))
12	11	sseq2d 3077	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑦 ⊆ (𝐵 ↾t 𝐴) ↔ 𝑦 ⊆ ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴))))
13	12	biimpa 292	. . . . . . . . . 10 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → 𝑦 ⊆ ran (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)))
14		vex 2644	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
15	14	inex1 4002	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝐴) ∈ V
16	15	rgenw 2446	. . . . . . . . . . 11 ⊢ ∀𝑥 ∈ 𝐵 (𝑥 ∩ 𝐴) ∈ V
17		eqid 2100	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) = (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴))
18	17	fnmpt 5185	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∩ 𝐴) ∈ V → (𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) Fn 𝐵)
19		fnima 5177	. . . . . . . . . . 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	syl6sseqr 3096	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → 𝑦 ⊆ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝐵))
22		ssimaexg 5415	. . . . . . . . 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 1184	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)))
24		df-ima 4490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ran ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧)
25		resmpt 4803	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧) = (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
26	25	adantl 273	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧) = (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
27	26	rneqd 4706	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ran ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) ↾ 𝑧) = ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
28	24, 27	syl5eq 2144	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
29	28	unieqd 3694	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ∪ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)))
30	15	dfiun3 4734	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴) = ∪ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴))
31	29, 30	syl6eqr 2150	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴))
32		iunin1 3824	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴) = (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴)
33	31, 32	syl6eq 2148	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) = (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴))
34		tgvalex 12001	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ∈ V)
35	34	ad2antrr 475	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (topGen‘𝐵) ∈ V)
36		simpr 109	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝐴 ∈ 𝑊)
37	36	adantr 272	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → 𝐴 ∈ 𝑊)
38		uniiun 3813	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 = ∪ 𝑥 ∈ 𝑧 𝑥
39		eltg3i 12007	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑧 ∈ (topGen‘𝐵))
40	38, 39	syl5eqelr 2187	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑥 ∈ 𝑧 𝑥 ∈ (topGen‘𝐵))
41	40	adantlr 464	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑥 ∈ 𝑧 𝑥 ∈ (topGen‘𝐵))
42		elrestr 11910	. . . . . . . . . . . . . 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 1184	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
44	33, 43	eqeltrd 2176	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴))
45		unieq 3692	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) → ∪ 𝑦 = ∪ ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧))
46	45	eleq1d 2168	. . . . . . . . . . . 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 156	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
48	47	expimpd 358	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
49	48	exlimdv 1758	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑦 = ((𝑥 ∈ 𝐵 ↦ (𝑥 ∩ 𝐴)) “ 𝑧)) → ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
50	49	adantr 272	. . . . . . . 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 2162	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ ∪ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
53	51, 52	syl5ibrcom 156	. . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑦 ⊆ (𝐵 ↾t 𝐴)) → (𝑥 = ∪ 𝑦 → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
54	53	expimpd 358	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑦 ⊆ (𝐵 ↾t 𝐴) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
55	54	exlimdv 1758	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑦(𝑦 ⊆ (𝐵 ↾t 𝐴) ∧ 𝑥 = ∪ 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
56	7, 55	sylbid 149	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (topGen‘(𝐵 ↾t 𝐴)) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
57	56	ssrdv 3053	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (topGen‘(𝐵 ↾t 𝐴)) ⊆ ((topGen‘𝐵) ↾t 𝐴))
58		restval 11908	. . . 4 ⊢ (((topGen‘𝐵) ∈ V ∧ 𝐴 ∈ 𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)))
59	34, 36, 58	syl2an2r 565	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)))
60		eltg3 12008	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧)))
61	60	adantr 272	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧)))
62	38	ineq1i 3220	. . . . . . . . . . . 12 ⊢ (∪ 𝑧 ∩ 𝐴) = (∪ 𝑥 ∈ 𝑧 𝑥 ∩ 𝐴)
63	62, 32	eqtr4i 2123	. . . . . . . . . . 11 ⊢ (∪ 𝑧 ∩ 𝐴) = ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴)
64		simplll 503	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → 𝐵 ∈ 𝑉)
65		simpllr 504	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → 𝐴 ∈ 𝑊)
66		simpr 109	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → 𝑧 ⊆ 𝐵)
67	66	sselda 3047	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝐵)
68		elrestr 11910	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
69	64, 65, 67, 68	syl3anc 1184	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) ∧ 𝑥 ∈ 𝑧) → (𝑥 ∩ 𝐴) ∈ (𝐵 ↾t 𝐴))
70	69	fmpttd 5507	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)):𝑧⟶(𝐵 ↾t 𝐴))
71	70	frnd 5218	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)) ⊆ (𝐵 ↾t 𝐴))
72		eltg3i 12007	. . . . . . . . . . . . 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 565	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ ran (𝑥 ∈ 𝑧 ↦ (𝑥 ∩ 𝐴)) ∈ (topGen‘(𝐵 ↾t 𝐴)))
74	30, 73	syl5eqel 2186	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → ∪ 𝑥 ∈ 𝑧 (𝑥 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)))
75	63, 74	syl5eqel 2186	. . . . . . . . . 10 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (∪ 𝑧 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)))
76		ineq1 3217	. . . . . . . . . . 11 ⊢ (𝑤 = ∪ 𝑧 → (𝑤 ∩ 𝐴) = (∪ 𝑧 ∩ 𝐴))
77	76	eleq1d 2168	. . . . . . . . . 10 ⊢ (𝑤 = ∪ 𝑧 → ((𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)) ↔ (∪ 𝑧 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
78	75, 77	syl5ibrcom 156	. . . . . . . . 9 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑧 ⊆ 𝐵) → (𝑤 = ∪ 𝑧 → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
79	78	expimpd 358	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
80	79	exlimdv 1758	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑤 = ∪ 𝑧) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
81	61, 80	sylbid 149	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑤 ∈ (topGen‘𝐵) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴))))
82	81	imp 123	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) ∧ 𝑤 ∈ (topGen‘𝐵)) → (𝑤 ∩ 𝐴) ∈ (topGen‘(𝐵 ↾t 𝐴)))
83	82	fmpttd 5507	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)):(topGen‘𝐵)⟶(topGen‘(𝐵 ↾t 𝐴)))
84	83	frnd 5218	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤 ∩ 𝐴)) ⊆ (topGen‘(𝐵 ↾t 𝐴)))
85	59, 84	eqsstrd 3083	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((topGen‘𝐵) ↾t 𝐴) ⊆ (topGen‘(𝐵 ↾t 𝐴)))
86	57, 85	eqssd 3064	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (topGen‘(𝐵 ↾t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1299  ∃wex 1436   ∈ wcel 1448  ∀wral 2375  Vcvv 2641   ∩ cin 3020   ⊆ wss 3021  ∪ cuni 3683  ∪ ciun 3760   ↦ cmpt 3929   × cxp 4475  ran crn 4478   ↾ cres 4479   “ cima 4480  Fun wfun 5053   Fn wfn 5054  ‘cfv 5059  (class class class)co 5706   ↾_t crest 11902  topGenctg 11917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-rest 11904  df-topgen 11923

This theorem is referenced by:  resttop  12121  txrest  12226