Theorem qtoptop2 23584

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 2729	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	qtopres 23583	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
3	2	3ad2ant2 1134	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
4		simp1 1136	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → 𝐽 ∈ Top)
5		funres 6524	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ ∪ 𝐽))
6	5	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → Fun (𝐹 ↾ ∪ 𝐽))
7		funforn 6743	. . . . . . . . . . . . . 14 ⊢ (Fun (𝐹 ↾ ∪ 𝐽) ↔ (𝐹 ↾ ∪ 𝐽):dom (𝐹 ↾ ∪ 𝐽)–onto→ran (𝐹 ↾ ∪ 𝐽))
8	6, 7	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐹 ↾ ∪ 𝐽):dom (𝐹 ↾ ∪ 𝐽)–onto→ran (𝐹 ↾ ∪ 𝐽))
9		dmres 5963	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹 ↾ ∪ 𝐽) = (∪ 𝐽 ∩ dom 𝐹)
10		inss1 4188	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝐽 ∩ dom 𝐹) ⊆ ∪ 𝐽
11	9, 10	eqsstri 3982	. . . . . . . . . . . . . 14 ⊢ dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽)
13	1	elqtop 23582	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (𝑦 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)))
15	14	simprbda 498	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → 𝑦 ⊆ ran (𝐹 ↾ ∪ 𝐽))
16		velpw 4556	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 ran (𝐹 ↾ ∪ 𝐽) ↔ 𝑦 ⊆ ran (𝐹 ↾ ∪ 𝐽))
17	15, 16	sylibr 234	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → 𝑦 ∈ 𝒫 ran (𝐹 ↾ ∪ 𝐽))
18	17	ex 412	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → 𝑦 ∈ 𝒫 ran (𝐹 ↾ ∪ 𝐽)))
19	18	ssrdv 3941	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽))
20		sstr2 3942	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ((𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽) → 𝑥 ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽)))
21	19, 20	syl5com 31	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → 𝑥 ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽)))
22		sspwuni 5049	. . . . . . 7 ⊢ (𝑥 ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽) ↔ ∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽))
23	21, 22	imbitrdi 251	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽)))
24		imauni 7182	. . . . . . . 8 ⊢ (◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) = ∪ 𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)
25	14	simplbda 499	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
26	25	ralrimiva 3121	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ∀𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))(◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
27		ssralv 4004	. . . . . . . . . 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 506	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → ∀𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
29		iunopn 22783	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽) → ∪ 𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
30	4, 28, 29	syl2an2r 685	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → ∪ 𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
31	24, 30	eqeltrid 2832	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → (◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽)
32	31	ex 412	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → (◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽))
33	23, 32	jcad 512	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → (∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽)))
34	1	elqtop 23582	. . . . . 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 1373	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽)))
36	33, 35	sylibrd 259	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))))
37	36	alrimiv 1927	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))))
38		inss1 4188	. . . . . 6 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
39	1	elqtop 23582	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽)))
41	40	biimpa 476	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → (𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽))
42	41	adantrr 717	. . . . . . 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 494	. . . . . 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 3947	. . . . 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 480	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → Fun (𝐹 ↾ ∪ 𝐽))
46		inpreima 6998	. . . . . . 7 ⊢ (Fun (𝐹 ↾ ∪ 𝐽) → (◡(𝐹 ↾ ∪ 𝐽) “ (𝑥 ∩ 𝑦)) = ((◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∩ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)))
47	45, 46	syl 17	. . . . . 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 480	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → 𝐽 ∈ Top)
49	42	simprd 495	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽)
50	25	adantrl 716	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
51		inopn 22784	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽 ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽) → ((◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∩ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)) ∈ 𝐽)
52	48, 49, 50, 51	syl3anc 1373	. . . . . 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 2828	. . . . 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 23582	. . . . . . 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 1373	. . . . . 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 480	. . . . 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 713	. . . 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 3172	. . 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 7382	. . . 4 ⊢ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∈ V
60		istopg 22780	. . . 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 583	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∈ Top)
63	3, 62	eqeltrd 2828	1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  ∪ ciun 4941  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   “ cima 5622  Fun wfun 6476  –onto→wfo 6480  (class class class)co 7349   qTop cqtop 17407  Topctop 22778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-qtop 17411  df-top 22779

This theorem is referenced by:  qtoptop  23585