Theorem qtopeu 22973

Description: Universal property of the quotient topology. If 𝐺 is a function from 𝐽 to 𝐾 which is equal on all equivalent elements under 𝐹, then there is a unique continuous map 𝑓:(𝐽 / 𝐹)⟶𝐾 such that 𝐺 = 𝑓 ∘ 𝐹, and we say that 𝐺 "passes to the quotient". (Contributed by Mario Carneiro, 24-Mar-2015.)

Hypotheses

Ref	Expression
qtopeu.1	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
qtopeu.3	⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
qtopeu.4	⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
qtopeu.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))

Assertion

Ref	Expression
qtopeu	⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓 ∘ 𝐹))

Distinct variable groups:   𝑥,𝑓,𝑦,𝐹   𝑓,𝐽,𝑥   𝑓,𝐾,𝑥   𝑥,𝑋,𝑦   𝑓,𝐺,𝑥,𝑦   𝜑,𝑓,𝑥,𝑦   𝑓,𝑌,𝑥

Allowed substitution hints:   𝐽(𝑦)   𝐾(𝑦)   𝑋(𝑓)   𝑌(𝑦)

Proof of Theorem qtopeu

Dummy variables 𝑔 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qtopeu.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
2		fofn 6746	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
3	1, 2	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝑋)
4	3	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹 Fn 𝑋)
5		fniniseg 6998	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝑋 → (𝑦 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = (𝐹‘𝑥))))
6	4, 5	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = (𝐹‘𝑥))))
7		eqcom 2744	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))
8	7	3anbi3i 1159	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = (𝐹‘𝑥)))
9		3anass 1095	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = (𝐹‘𝑥))))
10	8, 9	bitri 275	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = (𝐹‘𝑥))))
11		qtopeu.5	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))
12	10, 11	sylan2br 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = (𝐹‘𝑥)))) → (𝐺‘𝑥) = (𝐺‘𝑦))
13	12	eqcomd 2743	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = (𝐹‘𝑥)))) → (𝐺‘𝑦) = (𝐺‘𝑥))
14	13	expr 458	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = (𝐹‘𝑥)) → (𝐺‘𝑦) = (𝐺‘𝑥)))
15	6, 14	sylbid 239	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) → (𝐺‘𝑦) = (𝐺‘𝑥)))
16	15	ralrimiv 3139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ (◡𝐹 “ {(𝐹‘𝑥)})(𝐺‘𝑦) = (𝐺‘𝑥))
17		qtopeu.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
18		qtopeu.4	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
19		cntop2 22498	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
20	18, 19	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ Top)
21		toptopon2 22173	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
22	20, 21	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
23		cnf2 22506	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐺 ∈ (𝐽 Cn 𝐾)) → 𝐺:𝑋⟶∪ 𝐾)
24	17, 22, 18, 23	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:𝑋⟶∪ 𝐾)
25	24	ffnd 6657	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 Fn 𝑋)
26	25	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐺 Fn 𝑋)
27		cnvimass 6024	. . . . . . . . . . . . 13 ⊢ (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ dom 𝐹
28		fof 6744	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
29	1, 28	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
30	29	fdmd 6667	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐹 = 𝑋)
31	30	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → dom 𝐹 = 𝑋)
32	27, 31	sseqtrid 3988	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝑋)
33		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐺‘𝑦) → (𝑤 = (𝐺‘𝑥) ↔ (𝐺‘𝑦) = (𝐺‘𝑥)))
34	33	ralima 7175	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn 𝑋 ∧ (◡𝐹 “ {(𝐹‘𝑥)}) ⊆ 𝑋) → (∀𝑤 ∈ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)}))𝑤 = (𝐺‘𝑥) ↔ ∀𝑦 ∈ (◡𝐹 “ {(𝐹‘𝑥)})(𝐺‘𝑦) = (𝐺‘𝑥)))
35	26, 32, 34	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∀𝑤 ∈ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)}))𝑤 = (𝐺‘𝑥) ↔ ∀𝑦 ∈ (◡𝐹 “ {(𝐹‘𝑥)})(𝐺‘𝑦) = (𝐺‘𝑥)))
36	16, 35	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑤 ∈ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)}))𝑤 = (𝐺‘𝑥))
37	24	fdmd 6667	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐺 = 𝑋)
38	37	eleq2d 2823	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ dom 𝐺 ↔ 𝑥 ∈ 𝑋))
39	38	biimpar 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ dom 𝐺)
40		simpr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
41		eqidd 2738	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑥))
42		fniniseg 6998	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝑋 → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑥))))
43	4, 42	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑥))))
44	40, 41, 43	mpbir2and 711	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)}))
45		inelcm 4416	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ dom 𝐺 ∧ 𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑥)})) → (dom 𝐺 ∩ (◡𝐹 “ {(𝐹‘𝑥)})) ≠ ∅)
46	39, 44, 45	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (dom 𝐺 ∩ (◡𝐹 “ {(𝐹‘𝑥)})) ≠ ∅)
47		imadisj 6023	. . . . . . . . . . . . 13 ⊢ ((𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) = ∅ ↔ (dom 𝐺 ∩ (◡𝐹 “ {(𝐹‘𝑥)})) = ∅)
48	47	necon3bii 2994	. . . . . . . . . . . 12 ⊢ ((𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) ≠ ∅ ↔ (dom 𝐺 ∩ (◡𝐹 “ {(𝐹‘𝑥)})) ≠ ∅)
49	46, 48	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) ≠ ∅)
50		eqsn 4781	. . . . . . . . . . 11 ⊢ ((𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) ≠ ∅ → ((𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) = {(𝐺‘𝑥)} ↔ ∀𝑤 ∈ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)}))𝑤 = (𝐺‘𝑥)))
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) = {(𝐺‘𝑥)} ↔ ∀𝑤 ∈ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)}))𝑤 = (𝐺‘𝑥)))
52	36, 51	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) = {(𝐺‘𝑥)})
53	52	unieqd 4871	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) = ∪ {(𝐺‘𝑥)})
54		fvex 6843	. . . . . . . . 9 ⊢ (𝐺‘𝑥) ∈ V
55	54	unisn 4879	. . . . . . . 8 ⊢ ∪ {(𝐺‘𝑥)} = (𝐺‘𝑥)
56	53, 55	eqtr2di 2794	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) = ∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})))
57	56	mpteq2dva 5197	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐺‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)}))))
58	24	feqmptd 6898	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑋 ↦ (𝐺‘𝑥)))
59	29	ffvelcdmda 7022	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝑌)
60	29	feqmptd 6898	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝑥)))
61		eqidd 2738	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) = (𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))))
62		sneq 4588	. . . . . . . . . 10 ⊢ (𝑤 = (𝐹‘𝑥) → {𝑤} = {(𝐹‘𝑥)})
63	62	imaeq2d 6004	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑥) → (◡𝐹 “ {𝑤}) = (◡𝐹 “ {(𝐹‘𝑥)}))
64	63	imaeq2d 6004	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑥) → (𝐺 “ (◡𝐹 “ {𝑤})) = (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})))
65	64	unieqd 4871	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑥) → ∪ (𝐺 “ (◡𝐹 “ {𝑤})) = ∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})))
66	59, 60, 61, 65	fmptco 7062	. . . . . 6 ⊢ (𝜑 → ((𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ ∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)}))))
67	57, 58, 66	3eqtr4d 2787	. . . . 5 ⊢ (𝜑 → 𝐺 = ((𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) ∘ 𝐹))
68	67, 18	eqeltrrd 2839	. . . 4 ⊢ (𝜑 → ((𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) ∘ 𝐹) ∈ (𝐽 Cn 𝐾))
69	24	ffvelcdmda 7022	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ∪ 𝐾)
70	56, 69	eqeltrrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) ∈ ∪ 𝐾)
71	70	ralrimiva 3140	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) ∈ ∪ 𝐾)
72	65	eqcomd 2743	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹‘𝑥) → ∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) = ∪ (𝐺 “ (◡𝐹 “ {𝑤})))
73	72	eqcoms 2745	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = 𝑤 → ∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) = ∪ (𝐺 “ (◡𝐹 “ {𝑤})))
74	73	eleq1d 2822	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑤 → (∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) ∈ ∪ 𝐾 ↔ ∪ (𝐺 “ (◡𝐹 “ {𝑤})) ∈ ∪ 𝐾))
75	74	cbvfo 7222	. . . . . . . 8 ⊢ (𝐹:𝑋–onto→𝑌 → (∀𝑥 ∈ 𝑋 ∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) ∈ ∪ 𝐾 ↔ ∀𝑤 ∈ 𝑌 ∪ (𝐺 “ (◡𝐹 “ {𝑤})) ∈ ∪ 𝐾))
76	1, 75	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∪ (𝐺 “ (◡𝐹 “ {(𝐹‘𝑥)})) ∈ ∪ 𝐾 ↔ ∀𝑤 ∈ 𝑌 ∪ (𝐺 “ (◡𝐹 “ {𝑤})) ∈ ∪ 𝐾))
77	71, 76	mpbid 231	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ 𝑌 ∪ (𝐺 “ (◡𝐹 “ {𝑤})) ∈ ∪ 𝐾)
78		eqid 2737	. . . . . . 7 ⊢ (𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) = (𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤})))
79	78	fmpt 7045	. . . . . 6 ⊢ (∀𝑤 ∈ 𝑌 ∪ (𝐺 “ (◡𝐹 “ {𝑤})) ∈ ∪ 𝐾 ↔ (𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))):𝑌⟶∪ 𝐾)
80	77, 79	sylib 217	. . . . 5 ⊢ (𝜑 → (𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))):𝑌⟶∪ 𝐾)
81		qtopcn 22971	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) ∧ (𝐹:𝑋–onto→𝑌 ∧ (𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))):𝑌⟶∪ 𝐾)) → ((𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ ((𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) ∘ 𝐹) ∈ (𝐽 Cn 𝐾)))
82	17, 22, 1, 80, 81	syl22anc 837	. . . 4 ⊢ (𝜑 → ((𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ ((𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) ∘ 𝐹) ∈ (𝐽 Cn 𝐾)))
83	68, 82	mpbird 257	. . 3 ⊢ (𝜑 → (𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
84		coeq1 5804	. . . 4 ⊢ (𝑓 = (𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) → (𝑓 ∘ 𝐹) = ((𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) ∘ 𝐹))
85	84	rspceeqv 3588	. . 3 ⊢ (((𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝐺 = ((𝑤 ∈ 𝑌 ↦ ∪ (𝐺 “ (◡𝐹 “ {𝑤}))) ∘ 𝐹)) → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓 ∘ 𝐹))
86	83, 67, 85	syl2anc 585	. 2 ⊢ (𝜑 → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓 ∘ 𝐹))
87		eqtr2 2761	. . . 4 ⊢ ((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) → (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
88	1	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝐹:𝑋–onto→𝑌)
89		qtoptopon 22961	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
90	17, 1, 89	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
91	90	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
92	22	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝐾 ∈ (TopOn‘∪ 𝐾))
93		simprl 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
94		cnf2 22506	. . . . . . 7 ⊢ (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)) → 𝑓:𝑌⟶∪ 𝐾)
95	91, 92, 93, 94	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓:𝑌⟶∪ 𝐾)
96	95	ffnd 6657	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓 Fn 𝑌)
97		simprr 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
98		cnf2 22506	. . . . . . 7 ⊢ (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)) → 𝑔:𝑌⟶∪ 𝐾)
99	91, 92, 97, 98	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔:𝑌⟶∪ 𝐾)
100	99	ffnd 6657	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔 Fn 𝑌)
101		cocan2 7225	. . . . 5 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑓 Fn 𝑌 ∧ 𝑔 Fn 𝑌) → ((𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹) ↔ 𝑓 = 𝑔))
102	88, 96, 100, 101	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → ((𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹) ↔ 𝑓 = 𝑔))
103	87, 102	syl5ib 244	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → ((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) → 𝑓 = 𝑔))
104	103	ralrimivva 3194	. 2 ⊢ (𝜑 → ∀𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)∀𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) → 𝑓 = 𝑔))
105		coeq1 5804	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
106	105	eqeq2d 2748	. . 3 ⊢ (𝑓 = 𝑔 → (𝐺 = (𝑓 ∘ 𝐹) ↔ 𝐺 = (𝑔 ∘ 𝐹)))
107	106	reu4 3681	. 2 ⊢ (∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓 ∘ 𝐹) ↔ (∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓 ∘ 𝐹) ∧ ∀𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)∀𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) → 𝑓 = 𝑔)))
108	86, 104, 107	sylanbrc 584	1 ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓 ∘ 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  ∃!wreu 3348   ∩ cin 3901   ⊆ wss 3902  ∅c0 4274  {csn 4578  ∪ cuni 4857   ↦ cmpt 5180  ^◡ccnv 5624  dom cdm 5625   “ cima 5628   ∘ ccom 5629   Fn wfn 6479  ⟶wf 6480  –onto→wfo 6482  ‘cfv 6484  (class class class)co 7342   qTop cqtop 17312  Topctop 22148  TopOnctopon 22165   Cn ccn 22481

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-ov 7345  df-oprab 7346  df-mpo 7347  df-map 8693  df-qtop 17316  df-top 22149  df-topon 22166  df-cn 22484

This theorem is referenced by:  qtophmeo  23074