Theorem qtoptop2 23824

Description: The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)

Assertion

Ref	Expression
qtoptop2	⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)

Proof of Theorem qtoptop2

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

Step	Hyp	Ref	Expression
1		eqid 2769	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	qtopres 23823	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
3	2	3ad2ant2 1150	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
4		simp1 1152	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → 𝐽 ∈ Top)
5		funres 6579	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ ∪ 𝐽))
6	5	3ad2ant3 1151	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → Fun (𝐹 ↾ ∪ 𝐽))
7		funforn 6800	. . . . . . . . . . . . . 14 ⊢ (Fun (𝐹 ↾ ∪ 𝐽) ↔ (𝐹 ↾ ∪ 𝐽):dom (𝐹 ↾ ∪ 𝐽)–onto→ran (𝐹 ↾ ∪ 𝐽))
8	6, 7	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐹 ↾ ∪ 𝐽):dom (𝐹 ↾ ∪ 𝐽)–onto→ran (𝐹 ↾ ∪ 𝐽))
9		dmres 6012	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹 ↾ ∪ 𝐽) = (∪ 𝐽 ∩ dom 𝐹)
10		inss1 4197	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝐽 ∩ dom 𝐹) ⊆ ∪ 𝐽
11	9, 10	eqsstri 3991	. . . . . . . . . . . . . 14 ⊢ dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽)
13	1	elqtop 23822	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ (𝐹 ↾ ∪ 𝐽):dom (𝐹 ↾ ∪ 𝐽)–onto→ran (𝐹 ↾ ∪ 𝐽) ∧ dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽) → (𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (𝑦 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)))
14	4, 8, 12, 13	syl3anc 1396	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (𝑦 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)))
15	14	simprbda 503	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → 𝑦 ⊆ ran (𝐹 ↾ ∪ 𝐽))
16		velpw 4572	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 ran (𝐹 ↾ ∪ 𝐽) ↔ 𝑦 ⊆ ran (𝐹 ↾ ∪ 𝐽))
17	15, 16	sylibr 237	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → 𝑦 ∈ 𝒫 ran (𝐹 ↾ ∪ 𝐽))
18	17	ex 417	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → 𝑦 ∈ 𝒫 ran (𝐹 ↾ ∪ 𝐽)))
19	18	ssrdv 3951	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽))
20		sstr2 3952	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ((𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽) → 𝑥 ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽)))
21	19, 20	syl5com 32	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → 𝑥 ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽)))
22		sspwuni 5070	. . . . . . 7 ⊢ (𝑥 ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽) ↔ ∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽))
23	21, 22	imbitrdi 254	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽)))
24		imauni 7245	. . . . . . . 8 ⊢ (◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) = ∪ 𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)
25	14	simplbda 504	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
26	25	ralrimiva 3163	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ∀𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))(◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
27		ssralv 4014	. . . . . . . . . 10 ⊢ (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → (∀𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))(◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽))
28	26, 27	mpan9 515	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → ∀𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
29		iunopn 23023	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽) → ∪ 𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
30	4, 28, 29	syl2an2r 697	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → ∪ 𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
31	24, 30	eqeltrid 2873	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → (◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽)
32	31	ex 417	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → (◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽))
33	23, 32	jcad 521	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → (∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽)))
34	1	elqtop 23822	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝐹 ↾ ∪ 𝐽):dom (𝐹 ↾ ∪ 𝐽)–onto→ran (𝐹 ↾ ∪ 𝐽) ∧ dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽) → (∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽)))
35	4, 8, 12, 34	syl3anc 1396	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽)))
36	33, 35	sylibrd 262	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))))
37	36	alrimiv 1954	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))))
38		inss1 4197	. . . . . 6 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
39	1	elqtop 23822	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝐹 ↾ ∪ 𝐽):dom (𝐹 ↾ ∪ 𝐽)–onto→ran (𝐹 ↾ ∪ 𝐽) ∧ dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽) → (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽)))
40	4, 8, 12, 39	syl3anc 1396	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽)))
41	40	biimpa 481	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → (𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽))
42	41	adantrr 729	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽))
43	42	simpld 499	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽))
44	38, 43	sstrid 3956	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (𝑥 ∩ 𝑦) ⊆ ran (𝐹 ↾ ∪ 𝐽))
45	6	adantr 485	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → Fun (𝐹 ↾ ∪ 𝐽))
46		inpreima 7060	. . . . . . 7 ⊢ (Fun (𝐹 ↾ ∪ 𝐽) → (◡(𝐹 ↾ ∪ 𝐽) “ (𝑥 ∩ 𝑦)) = ((◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∩ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)))
47	45, 46	syl 18	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (◡(𝐹 ↾ ∪ 𝐽) “ (𝑥 ∩ 𝑦)) = ((◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∩ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)))
48	4	adantr 485	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → 𝐽 ∈ Top)
49	42	simprd 500	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽)
50	25	adantrl 728	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
51		inopn 23024	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽 ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽) → ((◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∩ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)) ∈ 𝐽)
52	48, 49, 50, 51	syl3anc 1396	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → ((◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∩ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)) ∈ 𝐽)
53	47, 52	eqeltrd 2869	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (◡(𝐹 ↾ ∪ 𝐽) “ (𝑥 ∩ 𝑦)) ∈ 𝐽)
54	1	elqtop 23822	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝐹 ↾ ∪ 𝐽):dom (𝐹 ↾ ∪ 𝐽)–onto→ran (𝐹 ↾ ∪ 𝐽) ∧ dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽) → ((𝑥 ∩ 𝑦) ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ ((𝑥 ∩ 𝑦) ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ (𝑥 ∩ 𝑦)) ∈ 𝐽)))
55	4, 8, 12, 54	syl3anc 1396	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ((𝑥 ∩ 𝑦) ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ ((𝑥 ∩ 𝑦) ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ (𝑥 ∩ 𝑦)) ∈ 𝐽)))
56	55	adantr 485	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → ((𝑥 ∩ 𝑦) ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ ((𝑥 ∩ 𝑦) ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ (𝑥 ∩ 𝑦)) ∈ 𝐽)))
57	44, 53, 56	mpbir2and 725	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (𝑥 ∩ 𝑦) ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
58	57	ralrimivva 3214	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ∀𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))∀𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))(𝑥 ∩ 𝑦) ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
59		ovex 7444	. . . 4 ⊢ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∈ V
60		istopg 23020	. . . 4 ⊢ ((𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∈ V → ((𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) ∧ ∀𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))∀𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))(𝑥 ∩ 𝑦) ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))))
61	59, 60	ax-mp 5	. . 3 ⊢ ((𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) ∧ ∀𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))∀𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))(𝑥 ∩ 𝑦) ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))))
62	37, 58, 61	sylanbrc 594	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∈ Top)
63	3, 62	eqeltrd 2869	1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101  ∀wal 1565   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4567  ∪ cuni 4876  ∪ ciun 4960  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   ↾ cres 5664   “ cima 5665  Fun wfun 6531  –onto→wfo 6535  (class class class)co 7411   qTop cqtop 17556  Topctop 23018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-qtop 17560  df-top 23019

This theorem is referenced by:  qtoptop  23825