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Theorem qtoptop2 21502

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 2622	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	qtopres 21501	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
3	2	3ad2ant2 1083	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) = (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
4		simp1 1061	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → 𝐽 ∈ Top)
5		funres 5929	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ ∪ 𝐽))
6	5	3ad2ant3 1084	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → Fun (𝐹 ↾ ∪ 𝐽))
7		funforn 6122	. . . . . . . . . . . . . 14 ⊢ (Fun (𝐹 ↾ ∪ 𝐽) ↔ (𝐹 ↾ ∪ 𝐽):dom (𝐹 ↾ ∪ 𝐽)–onto→ran (𝐹 ↾ ∪ 𝐽))
8	6, 7	sylib 208	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐹 ↾ ∪ 𝐽):dom (𝐹 ↾ ∪ 𝐽)–onto→ran (𝐹 ↾ ∪ 𝐽))
9		dmres 5419	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹 ↾ ∪ 𝐽) = (∪ 𝐽 ∩ dom 𝐹)
10		inss1 3833	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝐽 ∩ dom 𝐹) ⊆ ∪ 𝐽
11	9, 10	eqsstri 3635	. . . . . . . . . . . . . 14 ⊢ dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → dom (𝐹 ↾ ∪ 𝐽) ⊆ ∪ 𝐽)
13	1	elqtop 21500	. . . . . . . . . . . . 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 1326	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (𝑦 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)))
15	14	simprbda 653	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → 𝑦 ⊆ ran (𝐹 ↾ ∪ 𝐽))
16		selpw 4165	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 ran (𝐹 ↾ ∪ 𝐽) ↔ 𝑦 ⊆ ran (𝐹 ↾ ∪ 𝐽))
17	15, 16	sylibr 224	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → 𝑦 ∈ 𝒫 ran (𝐹 ↾ ∪ 𝐽))
18	17	ex 450	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → 𝑦 ∈ 𝒫 ran (𝐹 ↾ ∪ 𝐽)))
19	18	ssrdv 3609	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽))
20		sstr2 3610	. . . . . . . 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 4611	. . . . . . 7 ⊢ (𝑥 ⊆ 𝒫 ran (𝐹 ↾ ∪ 𝐽) ↔ ∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽))
23	21, 22	syl6ib 241	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽)))
24		imauni 6504	. . . . . . . 8 ⊢ (^◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) = ∪ 𝑦 ∈ 𝑥 (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)
25		simpl1 1064	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → 𝐽 ∈ Top)
26	14	simplbda 654	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
27	26	ralrimiva 2966	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ∀𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))(^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
28		ssralv 3666	. . . . . . . . . 10 ⊢ (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → (∀𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))(^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝑥 (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽))
29	27, 28	mpan9 486	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → ∀𝑦 ∈ 𝑥 (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
30		iunopn 20703	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑥 (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽) → ∪ 𝑦 ∈ 𝑥 (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
31	25, 29, 30	syl2anc 693	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → ∪ 𝑦 ∈ 𝑥 (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
32	24, 31	syl5eqel 2705	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → (^◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽)
33	32	ex 450	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → (^◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽))
34	23, 33	jcad 555	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → (∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (^◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽)))
35	1	elqtop 21500	. . . . . 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)))
36	4, 8, 12, 35	syl3anc 1326	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (∪ 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (^◡(𝐹 ↾ ∪ 𝐽) “ ∪ 𝑥) ∈ 𝐽)))
37	34, 36	sylibrd 249	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))))
38	37	alrimiv 1855	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ∀𝑥(𝑥 ⊆ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) → ∪ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))))
39		inss1 3833	. . . . . 6 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
40	1	elqtop 21500	. . . . . . . . . 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)))
41	4, 8, 12, 40	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ (𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽)))
42	41	biimpa 501	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ 𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))) → (𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽))
43	42	adantrr 753	. . . . . . 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))
44	43	simpld 475	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → 𝑥 ⊆ ran (𝐹 ↾ ∪ 𝐽))
45	39, 44	syl5ss 3614	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (𝑥 ∩ 𝑦) ⊆ ran (𝐹 ↾ ∪ 𝐽))
46	6	adantr 481	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → Fun (𝐹 ↾ ∪ 𝐽))
47		inpreima 6342	. . . . . . 7 ⊢ (Fun (𝐹 ↾ ∪ 𝐽) → (^◡(𝐹 ↾ ∪ 𝐽) “ (𝑥 ∩ 𝑦)) = ((^◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∩ (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)))
48	46, 47	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)))
49	4	adantr 481	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → 𝐽 ∈ Top)
50	43	simprd 479	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽)
51	26	adantrl 752	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽)
52		inopn 20704	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∈ 𝐽 ∧ (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦) ∈ 𝐽) → ((^◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∩ (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)) ∈ 𝐽)
53	49, 50, 51, 52	syl3anc 1326	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → ((^◡(𝐹 ↾ ∪ 𝐽) “ 𝑥) ∩ (^◡(𝐹 ↾ ∪ 𝐽) “ 𝑦)) ∈ 𝐽)
54	48, 53	eqeltrd 2701	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (^◡(𝐹 ↾ ∪ 𝐽) “ (𝑥 ∩ 𝑦)) ∈ 𝐽)
55	1	elqtop 21500	. . . . . . 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)))
56	4, 8, 12, 55	syl3anc 1326	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ((𝑥 ∩ 𝑦) ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ↔ ((𝑥 ∩ 𝑦) ⊆ ran (𝐹 ↾ ∪ 𝐽) ∧ (^◡(𝐹 ↾ ∪ 𝐽) “ (𝑥 ∩ 𝑦)) ∈ 𝐽)))
57	56	adantr 481	. . . . 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)))
58	45, 54, 57	mpbir2and 957	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) ∧ (𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∧ 𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))) → (𝑥 ∩ 𝑦) ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
59	58	ralrimivva 2971	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → ∀𝑥 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))∀𝑦 ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽))(𝑥 ∩ 𝑦) ∈ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)))
60		ovex 6678	. . . 4 ⊢ (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∈ V
61		istopg 20700	. . . 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)))))
62	60, 61	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))))
63	38, 59, 62	sylanbrc 698	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop (𝐹 ↾ ∪ 𝐽)) ∈ Top)
64	3, 63	eqeltrd 2701	1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 ∪ ciun 4520 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 Fun wfun 5882 –onto→wfo 5886 (class class class)co 6650 qTop cqtop 16163 Topctop 20698

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-qtop 16167 df-top 20699

This theorem is referenced by: qtoptop 21503

W3C validator