Theorem qtoprest 23604

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–onto→𝑌)
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–onto→𝑌)
3		fofn 6774	. . . . . . 7 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑋)
5		qtopid 23592	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
6	1, 4, 5	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
7		qtoprest.5	. . . . . . 7 ⊢ (𝜑 → 𝐴 = (◡𝐹 “ 𝑈))
8		cnvimass 6053	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑈) ⊆ dom 𝐹
9	4	fndmd 6623	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝑋)
10	8, 9	sseqtrid 3989	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ 𝑈) ⊆ 𝑋)
11	7, 10	eqsstrd 3981	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
12		toponuni 22801	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
13	1, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
14	11, 13	sseqtrd 3983	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
15		eqid 2729	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
16	15	cnrest 23172	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ 𝐴 ⊆ ∪ 𝐽) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐽 qTop 𝐹)))
17	6, 14, 16	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐽 qTop 𝐹)))
18		qtoptopon 23591	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
19	1, 2, 18	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
20		df-ima 5651	. . . . . . 7 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
21	7	imaeq2d 6031	. . . . . . . 8 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (◡𝐹 “ 𝑈)))
22		qtoprest.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝑌)
23		foimacnv 6817	. . . . . . . . 9 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑈 ⊆ 𝑌) → (𝐹 “ (◡𝐹 “ 𝑈)) = 𝑈)
24	2, 22, 23	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝑈)) = 𝑈)
25	21, 24	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ 𝐴) = 𝑈)
26	20, 25	eqtr3id 2778	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ 𝐴) = 𝑈)
27		eqimss 4005	. . . . . 6 ⊢ (ran (𝐹 ↾ 𝐴) = 𝑈 → ran (𝐹 ↾ 𝐴) ⊆ 𝑈)
28	26, 27	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ 𝐴) ⊆ 𝑈)
29		cnrest2 23173	. . . . 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 1373	. . . 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 232	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn ((𝐽 qTop 𝐹) ↾t 𝑈)))
32		resttopon 23048	. . . 4 ⊢ (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝑈 ⊆ 𝑌) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
33	19, 22, 32	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
34		qtopss 23602	. . 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 1373	. 2 ⊢ (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) ⊆ ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)))
36		resttopon 23048	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
37	1, 11, 36	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
38		fnfun 6618	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
39	4, 38	syl 17	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
40	11, 9	sseqtrrd 3984	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
41		fores 6782	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))
42	39, 40, 41	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))
43		foeq3 6770	. . . . . . 7 ⊢ ((𝐹 “ 𝐴) = 𝑈 → ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→𝑈))
44	25, 43	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–onto→𝑈))
45	42, 44	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–onto→𝑈)
46		elqtop3 23590	. . . . 5 ⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝑈) → (𝑥 ∈ ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)) ↔ (𝑥 ⊆ 𝑈 ∧ (◡(𝐹 ↾ 𝐴) “ 𝑥) ∈ (𝐽 ↾t 𝐴))))
47	37, 45, 46	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)) ↔ (𝑥 ⊆ 𝑈 ∧ (◡(𝐹 ↾ 𝐴) “ 𝑥) ∈ (𝐽 ↾t 𝐴))))
48		cnvresima 6203	. . . . . . . 8 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝑥) = ((◡𝐹 “ 𝑥) ∩ 𝐴)
49		imass2 6073	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑈 → (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝑈))
50	49	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝑈))
51	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → 𝐴 = (◡𝐹 “ 𝑈))
52	50, 51	sseqtrrd 3984	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → (◡𝐹 “ 𝑥) ⊆ 𝐴)
53		dfss2 3932	. . . . . . . . 9 ⊢ ((◡𝐹 “ 𝑥) ⊆ 𝐴 ↔ ((◡𝐹 “ 𝑥) ∩ 𝐴) = (◡𝐹 “ 𝑥))
54	52, 53	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → ((◡𝐹 “ 𝑥) ∩ 𝐴) = (◡𝐹 “ 𝑥))
55	48, 54	eqtrid 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → (◡(𝐹 ↾ 𝐴) “ 𝑥) = (◡𝐹 “ 𝑥))
56	55	eleq1d 2813	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → ((◡(𝐹 ↾ 𝐴) “ 𝑥) ∈ (𝐽 ↾t 𝐴) ↔ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴)))
57		simplrl 776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑥 ⊆ 𝑈)
58		dfss2 3932	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝑈 ↔ (𝑥 ∩ 𝑈) = 𝑥)
59	57, 58	sylib 218	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → (𝑥 ∩ 𝑈) = 𝑥)
60		topontop 22800	. . . . . . . . . . . 12 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → (𝐽 qTop 𝐹) ∈ Top)
61	19, 60	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Top)
62	61	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → (𝐽 qTop 𝐹) ∈ Top)
63		toponmax 22813	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
64	1, 63	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
65		focdmex 7934	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐽 → (𝐹:𝑋–onto→𝑌 → 𝑌 ∈ V))
66	64, 2, 65	sylc 65	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ V)
67	66, 22	ssexd 5279	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ V)
68	67	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑈 ∈ V)
69	22	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑈 ⊆ 𝑌)
70	57, 69	sstrd 3957	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑥 ⊆ 𝑌)
71		topontop 22800	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
72	1, 71	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ Top)
73		restopn2 23064	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → ((◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴) ↔ ((◡𝐹 “ 𝑥) ∈ 𝐽 ∧ (◡𝐹 “ 𝑥) ⊆ 𝐴)))
74	72, 73	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐽) → ((◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴) ↔ ((◡𝐹 “ 𝑥) ∈ 𝐽 ∧ (◡𝐹 “ 𝑥) ⊆ 𝐴)))
75	74	simprbda 498	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐽) ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴)) → (◡𝐹 “ 𝑥) ∈ 𝐽)
76	75	adantrl 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝐽) ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) → (◡𝐹 “ 𝑥) ∈ 𝐽)
77	76	an32s 652	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → (◡𝐹 “ 𝑥) ∈ 𝐽)
78		elqtop3 23590	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
79	1, 2, 78	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽)))
80	79	ad2antrr 726	. . . . . . . . . . 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 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑥 ∈ (𝐽 qTop 𝐹))
82		elrestr 17391	. . . . . . . . . 10 ⊢ (((𝐽 qTop 𝐹) ∈ Top ∧ 𝑈 ∈ V ∧ 𝑥 ∈ (𝐽 qTop 𝐹)) → (𝑥 ∩ 𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
83	62, 68, 81, 82	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → (𝑥 ∩ 𝑈) ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
84	59, 83	eqeltrrd 2829	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
85	33	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈))
86		toponuni 22801	. . . . . . . . . . . 12 ⊢ (((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈) → 𝑈 = ∪ ((𝐽 qTop 𝐹) ↾t 𝑈))
87	85, 86	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 = ∪ ((𝐽 qTop 𝐹) ↾t 𝑈))
88	87	difeq1d 4088	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈 ∖ 𝑥) = (∪ ((𝐽 qTop 𝐹) ↾t 𝑈) ∖ 𝑥))
89	22	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ 𝑌)
90	19	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
91		toponuni 22801	. . . . . . . . . . . . 13 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹))
92	90, 91	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑌 = ∪ (𝐽 qTop 𝐹))
93	89, 92	sseqtrd 3983	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ ∪ (𝐽 qTop 𝐹))
94	89	ssdifssd 4110	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈 ∖ 𝑥) ⊆ 𝑌)
95	39	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → Fun 𝐹)
96		funcnvcnv 6583	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
97		imadif 6600	. . . . . . . . . . . . . . 15 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑈 ∖ 𝑥)) = ((◡𝐹 “ 𝑈) ∖ (◡𝐹 “ 𝑥)))
98	95, 96, 97	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝑈 ∖ 𝑥)) = ((◡𝐹 “ 𝑈) ∖ (◡𝐹 “ 𝑥)))
99	7	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = (◡𝐹 “ 𝑈))
100	99	difeq1d 4088	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (◡𝐹 “ 𝑥)) = ((◡𝐹 “ 𝑈) ∖ (◡𝐹 “ 𝑥)))
101	98, 100	eqtr4d 2767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝑈 ∖ 𝑥)) = (𝐴 ∖ (◡𝐹 “ 𝑥)))
102		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ (Clsd‘𝐽))
103	37	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
104		toponuni 22801	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
105	103, 104	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
106	105	difeq1d 4088	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (◡𝐹 “ 𝑥)) = (∪ (𝐽 ↾t 𝐴) ∖ (◡𝐹 “ 𝑥)))
107		topontop 22800	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) → (𝐽 ↾t 𝐴) ∈ Top)
108	103, 107	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐴) ∈ Top)
109		simplrr 777	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))
110		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴)
111	110	opncld 22920	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ↾t 𝐴) ∈ Top ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴)) → (∪ (𝐽 ↾t 𝐴) ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
112	108, 109, 111	syl2anc 584	. . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
114		restcldr 23061	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘(𝐽 ↾t 𝐴))) → (𝐴 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘𝐽))
115	102, 113, 114	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘𝐽))
116	101, 115	eqeltrd 2828	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝑈 ∖ 𝑥)) ∈ (Clsd‘𝐽))
117		qtopcld 23600	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → ((𝑈 ∖ 𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈 ∖ 𝑥) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑈 ∖ 𝑥)) ∈ (Clsd‘𝐽))))
118	1, 2, 117	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑈 ∖ 𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)) ↔ ((𝑈 ∖ 𝑥) ⊆ 𝑌 ∧ (◡𝐹 “ (𝑈 ∖ 𝑥)) ∈ (Clsd‘𝐽))))
119	118	ad2antrr 726	. . . . . . . . . . . 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 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈 ∖ 𝑥) ∈ (Clsd‘(𝐽 qTop 𝐹)))
121		difssd 4100	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈 ∖ 𝑥) ⊆ 𝑈)
122		eqid 2729	. . . . . . . . . . . 12 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
123	122	restcldi 23060	. . . . . . . . . . 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 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈 ∖ 𝑥) ∈ (Clsd‘((𝐽 qTop 𝐹) ↾t 𝑈)))
125	88, 124	eqeltrrd 2829	. . . . . . . . 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 22800	. . . . . . . . . . 11 ⊢ (((𝐽 qTop 𝐹) ↾t 𝑈) ∈ (TopOn‘𝑈) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ Top)
127	85, 126	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → ((𝐽 qTop 𝐹) ↾t 𝑈) ∈ Top)
128		simplrl 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ 𝑈)
129	128, 87	sseqtrd 3983	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ↾t 𝑈))
130		eqid 2729	. . . . . . . . . . 11 ⊢ ∪ ((𝐽 qTop 𝐹) ↾t 𝑈) = ∪ ((𝐽 qTop 𝐹) ↾t 𝑈)
131	130	isopn2 22919	. . . . . . . . . 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 584	. . . . . . . . 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 257	. . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) → (𝐴 ∈ 𝐽 ∨ 𝐴 ∈ (Clsd‘𝐽)))
136	84, 133, 135	mpjaodan 960	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈))
137	136	expr 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → ((◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
138	56, 137	sylbid 240	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → ((◡(𝐹 ↾ 𝐴) “ 𝑥) ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
139	138	expimpd 453	. . . 4 ⊢ (𝜑 → ((𝑥 ⊆ 𝑈 ∧ (◡(𝐹 ↾ 𝐴) “ 𝑥) ∈ (𝐽 ↾t 𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
140	47, 139	sylbid 240	. . 3 ⊢ (𝜑 → (𝑥 ∈ ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)) → 𝑥 ∈ ((𝐽 qTop 𝐹) ↾t 𝑈)))
141	140	ssrdv 3952	. 2 ⊢ (𝜑 → ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)) ⊆ ((𝐽 qTop 𝐹) ↾t 𝑈))
142	35, 141	eqssd 3964	1 ⊢ (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∖ cdif 3911   ∩ cin 3913   ⊆ wss 3914  ∪ cuni 4871  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641  Fun wfun 6505   Fn wfn 6506  –onto→wfo 6509  ‘cfv 6511  (class class class)co 7387   ↾_t crest 17383   qTop cqtop 17466  Topctop 22780  TopOnctopon 22797  Clsdccld 22903   Cn ccn 23111

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-map 8801  df-en 8919  df-fin 8922  df-fi 9362  df-rest 17385  df-topgen 17406  df-qtop 17470  df-top 22781  df-topon 22798  df-bases 22833  df-cld 22906  df-cn 23114

This theorem is referenced by: (None)