Theorem qtoprest 24069

Description: If 𝐴 is a saturated open or closed set (where saturated means that 𝐴 = (^◡𝐹 “ 𝑈) for some 𝑈), then the restriction of the quotient map 𝐹 to 𝐴 is a quotient map. (Contributed by Mario Carneiro, 24-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
qtoprest.2	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
qtoprest.3	⊢ (𝜑 → 𝐹:𝑋⟶onto𝑌)
qtoprest.4	⊢ (𝜑 → 𝑈 ⊆ 𝑌)
qtoprest.5	⊢ (𝜑 → 𝐴 = (◡𝐹 “ 𝑈))
qtoprest.6	⊢ (𝜑 → (𝐴 ∈ 𝐽 ∨ 𝐴 ∈ (Clsd‘𝐽)))

Assertion

Ref	Expression
qtoprest	⊢ (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)))

Proof of Theorem qtoprest

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qtoprest.2	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		qtoprest.3	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶onto𝑌)
3		fofn 6823	. . . . . . 7 ⊢ (𝐹:𝑋⟶onto𝑌 → 𝐹 Fn 𝑋)
4	2, 3	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑋)
5		qtopid 24057	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
6	1, 4, 5	syl2anc 596	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7		qtoprest.5	. . . . . . 7 ⊢ (𝜑 → 𝐴 = (◡𝐹 “ 𝑈))
8		cnvimass 6212	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑈) ⊆ dom 𝐹
9	4	fndmd 6669	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝑋)
10	8, 9	sseqtrid 3955	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ 𝑈) ⊆ 𝑋)
11	7, 10	eqsstrd 3947	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
12		toponuni 23265	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
13	1, 12	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
14	11, 13	sseqtrd 3949	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
15		eqid 2763	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
16	15	cnrest 23636	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ 𝐴 ⊆ ∪ 𝐽) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐽 qTop 𝐹)))
17	6, 14, 16	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐽 qTop 𝐹)))
18		qtoptopon 24056	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
19	1, 2, 18	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
20		df-ima 5669	. . . . . . 7 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
21	7	imaeq2d 6065	. . . . . . . 8 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (◡𝐹 “ 𝑈)))
22		qtoprest.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝑌)
23		foimacnv 6867	. . . . . . . . 9 ⊢ ((𝐹:𝑋⟶onto𝑌 ∧ 𝑈 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑈)) = 𝑈)
24	2, 22, 23	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝑈)) = 𝑈)
25	21, 24	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ 𝐴) = 𝑈)
26	20, 25	eqtr3id 2812	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ 𝐴) = 𝑈)
27		eqimss 3971	. . . . . 6 ⊢ (ran (𝐹 ↾ 𝐴) = 𝑈 → ran (𝐹 ↾ 𝐴) ⊆ 𝑈)
28	26, 27	syl 18	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ 𝐴) ⊆ 𝑈)
29		cnrest2 23637	. . . . 5 ⊢ (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ ran (𝐹 ↾ 𝐴) ⊆ 𝑈 ∧ 𝑈 ⊆ 𝑌) → ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐽 qTop 𝐹)) ↔ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈))))
30	19, 28, 22, 29	syl3anc 1399	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐽 qTop 𝐹)) ↔ (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈))))
31	17, 30	mpbid 235	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈)))
32		resttopon 23512	. . . 4 ⊢ (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝑈 ⊆ 𝑌) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
33	19, 22, 32	syl2anc 596	. . 3 ⊢ (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
34		qtopss 24067	. . 3 ⊢ (((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈)) ∧ ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈) ∧ ran (𝐹 ↾ 𝐴) = 𝑈) → ((𝐽 qTop 𝐹) ↾t 𝑈) ⊆ ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)))
35	31, 33, 26, 34	syl3anc 1399	. 2 ⊢ (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ⊆ ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)))
36		resttopon 23512	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
37	1, 11, 36	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
38		fnfun 6664	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
39	4, 38	syl 18	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
40	11, 9	sseqtrrd 3950	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
41		fores 6831	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴⟶onto(𝐹 “ 𝐴))
42	39, 40, 41	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶onto(𝐹 “ 𝐴))
43		foeq3 6819	. . . . . . 7 ⊢ ((𝐹 “ 𝐴) = 𝑈 → ((𝐹 ↾ 𝐴):𝐴⟶onto(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴⟶onto𝑈))
44	25, 43	syl 18	. . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ 𝐴):𝐴⟶onto(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴⟶onto𝑈))
45	42, 44	mpbid 235	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶onto𝑈)
46		elqtop3 24055	. . . . 5 ⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐹 ↾ 𝐴):𝐴⟶onto𝑈) → (𝑥 ∈ ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)) ↔ (𝑥 ⊆ 𝑈 ∧ (◡(𝐹 ↾ 𝐴) “ 𝑥) ∈ (𝐽 ↾t 𝐴))))
47	37, 45, 46	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)) ↔ (𝑥 ⊆ 𝑈 ∧ (◡(𝐹 ↾ 𝐴) “ 𝑥) ∈ (𝐽 ↾t 𝐴))))
48		cnvresima 6245	. . . . . . . 8 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝑥) = ((◡𝐹 “ 𝑥) ∩ 𝐴)
49		imass2 6068	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑈 → (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝑈))
50	49	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝑈))
51	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → 𝐴 = (◡𝐹 “ 𝑈))
52	50, 51	sseqtrrd 3950	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → (◡𝐹 “ 𝑥) ⊆ 𝐴)
53		dfss2 4097	. . . . . . . . 9 ⊢ ((◡𝐹 “ 𝑥) ⊆ 𝐴 ↔ ((◡𝐹 “ 𝑥) ∩ 𝐴) = (◡𝐹 “ 𝑥))
54	52, 53	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → ((◡𝐹 “ 𝑥) ∩ 𝐴) = (◡𝐹 “ 𝑥))
55	48, 54	eqtrid 2810	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → (◡(𝐹 ↾ 𝐴) “ 𝑥) = (◡𝐹 “ 𝑥))
56	55	eleq1d 2849	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → ((◡(𝐹 ↾ 𝐴) “ 𝑥) ∈ (𝐽 ↾t 𝐴) ↔ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴)))
57		simplrl 789	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑥 ⊆ 𝑈)
58		dfss2 4097	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝑈 ↔ (𝑥 ∩ 𝑈) = 𝑥)
59	57, 58	sylib 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → (𝑥 ∩ 𝑈) = 𝑥)
60		topontop 23264	. . . . . . . . . . . 12 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
61	19, 60	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
62	61	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → (𝐽 qTop 𝐹) ∈ Top)
63		toponmax 23277	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
64	1, 63	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
65		focdmex 7995	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐽 → (𝐹:𝑋⟶onto𝑌 → 𝑌 ∈ V))
66	64, 2, 65	sylc 66	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ V)
67	66, 22	ssexd 5291	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ V)
68	67	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑈 ∈ V)
69	22	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑈 ⊆ 𝑌)
70	57, 69	sstrd 3923	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑥 ⊆ 𝑌)
71		topontop 23264	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
72	1, 71	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ Top)
73		restopn2 23528	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → ((◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴) ↔ ((◡𝐹 “ 𝑥) ∈ 𝐽 ∧ (◡𝐹 “ 𝑥) ⊆ 𝐴)))
74	72, 73	sylan 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐽) → ((◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴) ↔ ((◡𝐹 “ 𝑥) ∈ 𝐽 ∧ (◡𝐹 “ 𝑥) ⊆ 𝐴)))
75	74	simprbda 504	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐽) ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴)) → (◡𝐹 “ 𝑥) ∈ 𝐽)
76	75	adantrl 729	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐽) ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) → (◡𝐹 “ 𝑥) ∈ 𝐽)
77	76	an32s 665	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → (◡𝐹 “ 𝑥) ∈ 𝐽)
78		elqtop3 24055	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
79	1, 2, 78	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
80	79	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
81	70, 77, 80	mpbir2and 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑥 ∈ (𝐽 qTop 𝐹))
82		elrestr 17631	. . . . . . . . . 10 ⊢ (((𝐽 qTop 𝐹) ∈ Top ∧ 𝑈 ∈ V ∧ 𝑥 ∈ (𝐽 qTop 𝐹)) → (𝑥 ∩ 𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
83	62, 68, 81, 82	syl3anc 1399	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → (𝑥 ∩ 𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
84	59, 83	eqeltrrd 2865	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
85	33	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
86		toponuni 23265	. . . . . . . . . . . 12 ⊢ (((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈) → 𝑈 = ∪ ((𝐽 qTop 𝐹) ↾t 𝑈))
87	85, 86	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 = ∪ ((𝐽 qTop 𝐹) ↾t 𝑈))
88	87	difeq1d 4065	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈 ∖ 𝑥) = (∪ ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥))
89	22	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ 𝑌)
90	19	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
91		toponuni 23265	. . . . . . . . . . . . 13 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
92	90, 91	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑌 = ∪ (𝐽 qTop 𝐹))
93	89, 92	sseqtrd 3949	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ ∪ (𝐽 qTop 𝐹))
94	89	ssdifssd 4086	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈 ∖ 𝑥) ⊆ 𝑌)
95	39	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → Fun 𝐹)
96		funcnvcnv 6632	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
97		imadif 6649	. . . . . . . . . . . . . . 15 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑈 ∖ 𝑥)) = ((◡𝐹 “ 𝑈) ∖ (◡𝐹 “ 𝑥)))
98	95, 96, 97	3syl 19	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝑈 ∖ 𝑥)) = ((◡𝐹 “ 𝑈) ∖ (◡𝐹 “ 𝑥)))
99	7	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = (◡𝐹 “ 𝑈))
100	99	difeq1d 4065	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (◡𝐹 “ 𝑥)) = ((◡𝐹 “ 𝑈) ∖ (◡𝐹 “ 𝑥)))
101	98, 100	eqtr4d 2801	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝑈 ∖ 𝑥)) = (𝐴 ∖ (◡𝐹 “ 𝑥)))
102		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))
103	37	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
104		toponuni 23265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
105	103, 104	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
106	105	difeq1d 4065	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (◡𝐹 “ 𝑥)) = (∪ (𝐽 ↾t 𝐴) ∖ (◡𝐹 “ 𝑥)))
107		topontop 23264	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) → (𝐽 ↾t 𝐴) ∈ Top)
108	103, 107	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ Top)
109		simplrr 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))
110		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴)
111	110	opncld 23384	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ↾t 𝐴) ∈ Top ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴)) → (∪ (𝐽 ↾t 𝐴) ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
112	108, 109, 111	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (∪ (𝐽 ↾t 𝐴) ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
113	106, 112	eqeltrd 2864	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
114		restcldr 23525	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘(𝐽 ↾t 𝐴))) → (𝐴 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘𝐽))
115	102, 113, 114	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘𝐽))
116	101, 115	eqeltrd 2864	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝑈 ∖ 𝑥)) ∈ (Clsd‘𝐽))
117		qtopcld 24065	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶onto𝑌) → ((𝑈 ∖ 𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈 ∖ 𝑥) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑈 ∖ 𝑥)) ∈ (Clsd‘𝐽))))
118	1, 2, 117	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑈 ∖ 𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈 ∖ 𝑥) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑈 ∖ 𝑥)) ∈ (Clsd‘𝐽))))
119	118	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝑈 ∖ 𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈 ∖ 𝑥) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑈 ∖ 𝑥)) ∈ (Clsd‘𝐽))))
120	94, 116, 119	mpbir2and 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈 ∖ 𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)))
121		difssd 4076	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈 ∖ 𝑥) ⊆ 𝑈)
122		eqid 2763	. . . . . . . . . . . 12 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
123	122	restcldi 23524	. . . . . . . . . . 11 ⊢ ((𝑈 ⊆ ∪ (𝐽 qTop 𝐹) ∧ (𝑈 ∖ 𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ∧ (𝑈 ∖ 𝑥) ⊆ 𝑈) → (𝑈 ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
124	93, 120, 121, 123	syl3anc 1399	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈 ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
125	88, 124	eqeltrrd 2865	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (∪ ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
126		topontop 23264	. . . . . . . . . . 11 ⊢ (((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ Top)
127	85, 126	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ Top)
128		simplrl 789	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ 𝑈)
129	128, 87	sseqtrd 3949	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ↾t 𝑈))
130		eqid 2763	. . . . . . . . . . 11 ⊢ ∪ ((𝐽 qTop 𝐹) ↾t 𝑈) = ∪ ((𝐽 qTop 𝐹) ↾t 𝑈)
131	130	isopn2 23383	. . . . . . . . . 10 ⊢ ((((𝐽 qTop 𝐹) ↾t 𝑈) ∈ Top ∧ 𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ↾t 𝑈)) → (𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈) ↔ (∪ ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈))))
132	127, 129, 131	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈) ↔ (∪ ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈))))
133	125, 132	mpbird 260	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
134		qtoprest.6	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ 𝐽 ∨ 𝐴 ∈ (Clsd‘𝐽)))
135	134	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) → (𝐴 ∈ 𝐽 ∨ 𝐴 ∈ (Clsd‘𝐽)))
136	84, 133, 135	mpjaodan 973	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
137	136	expr 462	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → ((◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
138	56, 137	sylbid 243	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → ((◡(𝐹 ↾ 𝐴) “ 𝑥) ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
139	138	expimpd 459	. . . 4 ⊢ (𝜑 → ((𝑥 ⊆ 𝑈 ∧ (◡(𝐹 ↾ 𝐴) “ 𝑥) ∈ (𝐽 ↾t 𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
140	47, 139	sylbid 243	. . 3 ⊢ (𝜑 → (𝑥 ∈ ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
141	140	ssrdv 3919	. 2 ⊢ (𝜑 → ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)) ⊆ ((𝐽 qTop 𝐹) ↾t 𝑈))
142	35, 141	eqssd 3930	1 ⊢ (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  ∪ cuni 4870  ^◡ccnv 5655  dom cdm 5656  ran crn 5657   ↾ cres 5658   “ cima 5659  Fun wfun 6555   Fn wfn 6556  ⟶ontowfo 6559  ‘cfv 6561  (class class class)co 7446   ↾_t crest 17623   qTop cqtop 17707  Topctop 23244  TopOnctopon 23261  Clsdccld 23367   Cn ccn 23575

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-map 8880  df-en 9005  df-fin 9008  df-fi 9433  df-rest 17625  df-topgen 17646  df-qtop 17711  df-top 23245  df-topon 23262  df-bases 23297  df-cld 23370  df-cn 23578

This theorem is used by: (None)