Theorem qtoptop2 23664

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 2736	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	qtopres 23663	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
3	2	3ad2ant2 1135	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
4		simp1 1137	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → 𝐽 ∈ Top)
5		funres 6540	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ ∪ 𝐽))
6	5	3ad2ant3 1136	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → Fun (𝐹 ↾ ∪ 𝐽))
7		funforn 6759	. . . . . . . . . . . . . 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 5977	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹 ↾ ∪ 𝐽) = (∪ 𝐽 ∩ dom 𝐹)
10		inss1 4177	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝐽 ∩ dom 𝐹) ⊆ ∪ 𝐽
11	9, 10	eqsstri 3968	. . . . . . . . . . . . . 14 ⊢ dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽)
13	1	elqtop 23662	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 4546	. . . . . . . . . . 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 3927	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽))
20		sstr2 3928	. . . . . . . 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 5042	. . . . . . 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 7201	. . . . . . . 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 3129	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ∀𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))(◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
27		ssralv 3990	. . . . . . . . . 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 22863	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽) → ∪ 𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
30	4, 28, 29	syl2an2r 686	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → ∪ 𝑦 ∈ 𝑥 (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
31	24, 30	eqeltrid 2840	. . . . . . 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 23662	. . . . . 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 1374	. . . . 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 1929	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))))
38		inss1 4177	. . . . . 6 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
39	1	elqtop 23662	. . . . . . . . . 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 1374	. . . . . . . . 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 718	. . . . . . 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 3933	. . . . 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 7016	. . . . . . 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 717	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
51		inopn 22864	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽 ∧ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽) → ((◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∩ (◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)) ∈ 𝐽)
52	48, 49, 50, 51	syl3anc 1374	. . . . . 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 2836	. . . . 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 23662	. . . . . . 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 1374	. . . . . 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 714	. . . 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 3180	. . 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 7400	. . . 4 ⊢ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∈ V
60		istopg 22860	. . . 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 584	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∈ Top)
63	3, 62	eqeltrd 2836	1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850  ∪ ciun 4933  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634  Fun wfun 6492  –onto→wfo 6496  (class class class)co 7367   qTop cqtop 17467  Topctop 22858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-qtop 17471  df-top 22859

This theorem is referenced by:  qtoptop  23665