Theorem qtoprest 23633

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 6742	. . . . . . 7 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑋)
5		qtopid 23621	. . . . . 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 6035	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑈) ⊆ dom 𝐹
9	4	fndmd 6591	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝑋)
10	8, 9	sseqtrid 3973	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ 𝑈) ⊆ 𝑋)
11	7, 10	eqsstrd 3965	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
12		toponuni 22830	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
13	1, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
14	11, 13	sseqtrd 3967	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
15		eqid 2733	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
16	15	cnrest 23201	. . . . 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 23620	. . . . . 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 5632	. . . . . . 7 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
21	7	imaeq2d 6013	. . . . . . . 8 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐹 “ (◡𝐹 “ 𝑈)))
22		qtoprest.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝑌)
23		foimacnv 6785	. . . . . . . . 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 2768	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ 𝐴) = 𝑈)
26	20, 25	eqtr3id 2782	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ 𝐴) = 𝑈)
27		eqimss 3989	. . . . . 6 ⊢ (ran (𝐹 ↾ 𝐴) = 𝑈 → ran (𝐹 ↾ 𝐴) ⊆ 𝑈)
28	26, 27	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ 𝐴) ⊆ 𝑈)
29		cnrest2 23202	. . . . 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 23077	. . . 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 23631	. . 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 23077	. . . . . 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 6586	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
39	4, 38	syl 17	. . . . . . 7 ⊢ (𝜑 → Fun 𝐹)
40	11, 9	sseqtrrd 3968	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
41		fores 6750	. . . . . . 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 6738	. . . . . . 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 23619	. . . . 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 6182	. . . . . . . 8 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝑥) = ((◡𝐹 “ 𝑥) ∩ 𝐴)
49		imass2 6055	. . . . . . . . . . 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 3968	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → (◡𝐹 “ 𝑥) ⊆ 𝐴)
53		dfss2 3916	. . . . . . . . 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 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑈) → (◡(𝐹 ↾ 𝐴) “ 𝑥) = (◡𝐹 “ 𝑥))
56	55	eleq1d 2818	. . . . . 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 3916	. . . . . . . . . 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 22829	. . . . . . . . . . . 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 22842	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
64	1, 63	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
65		focdmex 7894	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐽 → (𝐹:𝑋–onto→𝑌 → 𝑌 ∈ V))
66	64, 2, 65	sylc 65	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ V)
67	66, 22	ssexd 5264	. . . . . . . . . . 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 3941	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ 𝐽) → 𝑥 ⊆ 𝑌)
71		topontop 22829	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
72	1, 71	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ Top)
73		restopn2 23093	. . . . . . . . . . . . . . 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 23619	. . . . . . . . . . . . 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 17334	. . . . . . . . . 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 2834	. . . . . . . 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 22830	. . . . . . . . . . . 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 4074	. . . . . . . . . 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 22830	. . . . . . . . . . . . 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 3967	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ ∪ (𝐽 qTop 𝐹))
94	89	ssdifssd 4096	. . . . . . . . . . . 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 6553	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
97		imadif 6570	. . . . . . . . . . . . . . 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 4074	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (◡𝐹 “ 𝑥)) = ((◡𝐹 “ 𝑈) ∖ (◡𝐹 “ 𝑥)))
101	98, 100	eqtr4d 2771	. . . . . . . . . . . . 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 22830	. . . . . . . . . . . . . . . . 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 4074	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (◡𝐹 “ 𝑥)) = (∪ (𝐽 ↾t 𝐴) ∖ (◡𝐹 “ 𝑥)))
107		topontop 22829	. . . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝐽 ↾t 𝐴) = ∪ (𝐽 ↾t 𝐴)
111	110	opncld 22949	. . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∖ (◡𝐹 “ 𝑥)) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
114		restcldr 23090	. . . . . . . . . . . . . 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 2833	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡𝐹 “ (𝑈 ∖ 𝑥)) ∈ (Clsd‘𝐽))
117		qtopcld 23629	. . . . . . . . . . . . . 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 4086	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑈 ∖ 𝑥) ⊆ 𝑈)
122		eqid 2733	. . . . . . . . . . . 12 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹)
123	122	restcldi 23089	. . . . . . . . . . 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 2834	. . . . . . . . 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 22829	. . . . . . . . . . 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 3967	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ⊆ 𝑈 ∧ (◡𝐹 “ 𝑥) ∈ (𝐽 ↾t 𝐴))) ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ ∪ ((𝐽 qTop 𝐹) ↾t 𝑈))
130		eqid 2733	. . . . . . . . . . 11 ⊢ ∪ ((𝐽 qTop 𝐹) ↾t 𝑈) = ∪ ((𝐽 qTop 𝐹) ↾t 𝑈)
131	130	isopn2 22948	. . . . . . . . . 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 3936	. 2 ⊢ (𝜑 → ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)) ⊆ ((𝐽 qTop 𝐹) ↾t 𝑈))
142	35, 141	eqssd 3948	1 ⊢ (𝜑 → ((𝐽 qTop 𝐹) ↾t 𝑈) = ((𝐽 ↾t 𝐴) qTop (𝐹 ↾ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∪ cuni 4858  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   “ cima 5622  Fun wfun 6480   Fn wfn 6481  –onto→wfo 6484  ‘cfv 6486  (class class class)co 7352   ↾_t crest 17326   qTop cqtop 17409  Topctop 22809  TopOnctopon 22826  Clsdccld 22932   Cn ccn 23140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  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-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-map 8758  df-en 8876  df-fin 8879  df-fi 9302  df-rest 17328  df-topgen 17349  df-qtop 17413  df-top 22810  df-topon 22827  df-bases 22862  df-cld 22935  df-cn 23143

This theorem is referenced by: (None)