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Theorem qtophmeo 23168

Description: If two functions on a base topology 𝐽 make the same identifications in order to create quotient spaces 𝐽 qTop 𝐹 and 𝐽 qTop 𝐺, then not only are 𝐽 qTop 𝐹 and 𝐽 qTop 𝐺 homeomorphic, but there is a unique homeomorphism that makes the diagram commute. (Contributed by Mario Carneiro, 24-Mar-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

**Hypotheses**
Ref	Expression
qtophmeo.2	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
qtophmeo.3	⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
qtophmeo.4	⊢ (𝜑 → 𝐺:𝑋–onto→𝑌)
qtophmeo.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐺‘𝑥) = (𝐺‘𝑦)))

**Assertion**
Ref	Expression
qtophmeo	⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))

Distinct variable groups: 𝑥,𝑓,𝑦,𝐹 𝑓,𝐺,𝑥,𝑦 𝑓,𝐽,𝑥,𝑦 𝜑,𝑓,𝑥,𝑦 𝑥,𝑋,𝑦 𝑓,𝑌,𝑥 Allowed substitution hints: 𝑋(𝑓) 𝑌(𝑦)

Proof of Theorem qtophmeo Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qtophmeo.2	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		qtophmeo.3	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
3		qtophmeo.4	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝑋–onto→𝑌)
4		fofn 6758	. . . . . . 7 ⊢ (𝐺:𝑋–onto→𝑌 → 𝐺 Fn 𝑋)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝑋)
6		qtopid 23056	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺 Fn 𝑋) → 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
7	1, 5, 6	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
8		df-3an 1089	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)))
9		qtophmeo.5	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐺‘𝑥) = (𝐺‘𝑦)))
10	9	biimpd 228	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐺‘𝑥) = (𝐺‘𝑦)))
11	10	impr 455	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))
12	8, 11	sylan2b 594	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))
13	1, 2, 7, 12	qtopeu 23067	. . . 4 ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
14		reurex 3357	. . . 4 ⊢ (∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹) → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
15	13, 14	syl 17	. . 3 ⊢ (𝜑 → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
16		simprl 769	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)))
17		fofn 6758	. . . . . . . . . 10 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
18	2, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝑋)
19		qtopid 23056	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
20	1, 18, 19	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
21		df-3an 1089	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)))
22	9	biimprd 247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐺‘𝑥) = (𝐺‘𝑦) → (𝐹‘𝑥) = (𝐹‘𝑦)))
23	22	impr 455	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐺‘𝑥) = (𝐺‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
24	21, 23	sylan2b 594	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐺‘𝑥) = (𝐺‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
25	1, 3, 20, 24	qtopeu 23067	. . . . . . 7 ⊢ (𝜑 → ∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺))
26	25	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → ∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺))
27		reurex 3357	. . . . . 6 ⊢ (∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺) → ∃𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺))
28	26, 27	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → ∃𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺))
29		qtoptopon 23055	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
30	1, 2, 29	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
31	30	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
32		qtoptopon 23055	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺:𝑋–onto→𝑌) → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
33	1, 3, 32	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
34	33	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
35		simplrl 775	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)))
36		cnf2 22600	. . . . . . . 8 ⊢ (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))) → 𝑓:𝑌⟶𝑌)
37	31, 34, 35, 36	syl3anc 1371	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝑓:𝑌⟶𝑌)
38		simprl 769	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
39		cnf2 22600	. . . . . . . 8 ⊢ (((𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))) → 𝑔:𝑌⟶𝑌)
40	34, 31, 38, 39	syl3anc 1371	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝑔:𝑌⟶𝑌)
41		coeq1 5813	. . . . . . . . 9 ⊢ (ℎ = (𝑔 ∘ 𝑓) → (ℎ ∘ 𝐹) = ((𝑔 ∘ 𝑓) ∘ 𝐹))
42	41	eqeq2d 2747	. . . . . . . 8 ⊢ (ℎ = (𝑔 ∘ 𝑓) → (𝐹 = (ℎ ∘ 𝐹) ↔ 𝐹 = ((𝑔 ∘ 𝑓) ∘ 𝐹)))
43		coeq1 5813	. . . . . . . . 9 ⊢ (ℎ = ( I ↾ 𝑌) → (ℎ ∘ 𝐹) = (( I ↾ 𝑌) ∘ 𝐹))
44	43	eqeq2d 2747	. . . . . . . 8 ⊢ (ℎ = ( I ↾ 𝑌) → (𝐹 = (ℎ ∘ 𝐹) ↔ 𝐹 = (( I ↾ 𝑌) ∘ 𝐹)))
45		simpr3 1196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
46	1, 2, 20, 45	qtopeu 23067	. . . . . . . . 9 ⊢ (𝜑 → ∃!ℎ ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹))𝐹 = (ℎ ∘ 𝐹))
47	46	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → ∃!ℎ ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹))𝐹 = (ℎ ∘ 𝐹))
48		cnco 22617	. . . . . . . . 9 ⊢ ((𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))) → (𝑔 ∘ 𝑓) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
49	35, 38, 48	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → (𝑔 ∘ 𝑓) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
50		idcn 22608	. . . . . . . . . 10 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
51	30, 50	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
52	51	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
53		simprr 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝐹 = (𝑔 ∘ 𝐺))
54		simplrr 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝐺 = (𝑓 ∘ 𝐹))
55	54	coeq2d 5818	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → (𝑔 ∘ 𝐺) = (𝑔 ∘ (𝑓 ∘ 𝐹)))
56	53, 55	eqtrd 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝐹 = (𝑔 ∘ (𝑓 ∘ 𝐹)))
57		coass 6217	. . . . . . . . 9 ⊢ ((𝑔 ∘ 𝑓) ∘ 𝐹) = (𝑔 ∘ (𝑓 ∘ 𝐹))
58	56, 57	eqtr4di 2794	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝐹 = ((𝑔 ∘ 𝑓) ∘ 𝐹))
59		fof 6756	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
60	2, 59	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
61	60	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝐹:𝑋⟶𝑌)
62		fcoi2 6717	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → (( I ↾ 𝑌) ∘ 𝐹) = 𝐹)
63	61, 62	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → (( I ↾ 𝑌) ∘ 𝐹) = 𝐹)
64	63	eqcomd 2742	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝐹 = (( I ↾ 𝑌) ∘ 𝐹))
65	42, 44, 47, 49, 52, 58, 64	reu2eqd 3694	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → (𝑔 ∘ 𝑓) = ( I ↾ 𝑌))
66		coeq1 5813	. . . . . . . . 9 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ ∘ 𝐺) = ((𝑓 ∘ 𝑔) ∘ 𝐺))
67	66	eqeq2d 2747	. . . . . . . 8 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (𝐺 = (ℎ ∘ 𝐺) ↔ 𝐺 = ((𝑓 ∘ 𝑔) ∘ 𝐺)))
68		coeq1 5813	. . . . . . . . 9 ⊢ (ℎ = ( I ↾ 𝑌) → (ℎ ∘ 𝐺) = (( I ↾ 𝑌) ∘ 𝐺))
69	68	eqeq2d 2747	. . . . . . . 8 ⊢ (ℎ = ( I ↾ 𝑌) → (𝐺 = (ℎ ∘ 𝐺) ↔ 𝐺 = (( I ↾ 𝑌) ∘ 𝐺)))
70		simpr3 1196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐺‘𝑥) = (𝐺‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))
71	1, 3, 7, 70	qtopeu 23067	. . . . . . . . 9 ⊢ (𝜑 → ∃!ℎ ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺))𝐺 = (ℎ ∘ 𝐺))
72	71	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → ∃!ℎ ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺))𝐺 = (ℎ ∘ 𝐺))
73		cnco 22617	. . . . . . . . 9 ⊢ ((𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))) → (𝑓 ∘ 𝑔) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
74	38, 35, 73	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → (𝑓 ∘ 𝑔) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
75		idcn 22608	. . . . . . . . . 10 ⊢ ((𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
76	33, 75	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
77	76	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
78	53	coeq2d 5818	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → (𝑓 ∘ 𝐹) = (𝑓 ∘ (𝑔 ∘ 𝐺)))
79	54, 78	eqtrd 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝐺 = (𝑓 ∘ (𝑔 ∘ 𝐺)))
80		coass 6217	. . . . . . . . 9 ⊢ ((𝑓 ∘ 𝑔) ∘ 𝐺) = (𝑓 ∘ (𝑔 ∘ 𝐺))
81	79, 80	eqtr4di 2794	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝐺 = ((𝑓 ∘ 𝑔) ∘ 𝐺))
82		fof 6756	. . . . . . . . . . . 12 ⊢ (𝐺:𝑋–onto→𝑌 → 𝐺:𝑋⟶𝑌)
83	3, 82	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:𝑋⟶𝑌)
84	83	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝐺:𝑋⟶𝑌)
85		fcoi2 6717	. . . . . . . . . 10 ⊢ (𝐺:𝑋⟶𝑌 → (( I ↾ 𝑌) ∘ 𝐺) = 𝐺)
86	84, 85	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → (( I ↾ 𝑌) ∘ 𝐺) = 𝐺)
87	86	eqcomd 2742	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → 𝐺 = (( I ↾ 𝑌) ∘ 𝐺))
88	67, 69, 72, 74, 77, 81, 87	reu2eqd 3694	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → (𝑓 ∘ 𝑔) = ( I ↾ 𝑌))
89	37, 40, 65, 88	2fcoidinvd 7241	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → ^◡𝑓 = 𝑔)
90	89, 38	eqeltrd 2838	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) → ^◡𝑓 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
91	28, 90	rexlimddv 3158	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → ^◡𝑓 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
92		ishmeo 23110	. . . 4 ⊢ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ↔ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ ^◡𝑓 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))))
93	16, 91, 92	sylanbrc 583	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))
94		simprr 771	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → 𝐺 = (𝑓 ∘ 𝐹))
95	15, 93, 94	reximssdv 3169	. 2 ⊢ (𝜑 → ∃𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
96		eqtr2 2760	. . . 4 ⊢ ((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) → (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
97	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝐹:𝑋–onto→𝑌)
98	30	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
99	33	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
100		simprl 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))
101		hmeof1o2 23114	. . . . . . 7 ⊢ (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))) → 𝑓:𝑌–1-1-onto→𝑌)
102	98, 99, 100, 101	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑓:𝑌–1-1-onto→𝑌)
103		f1ofn 6785	. . . . . 6 ⊢ (𝑓:𝑌–1-1-onto→𝑌 → 𝑓 Fn 𝑌)
104	102, 103	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑓 Fn 𝑌)
105		simprr 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))
106		hmeof1o2 23114	. . . . . . 7 ⊢ (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))) → 𝑔:𝑌–1-1-onto→𝑌)
107	98, 99, 105, 106	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑔:𝑌–1-1-onto→𝑌)
108		f1ofn 6785	. . . . . 6 ⊢ (𝑔:𝑌–1-1-onto→𝑌 → 𝑔 Fn 𝑌)
109	107, 108	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑔 Fn 𝑌)
110		cocan2 7238	. . . . 5 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑓 Fn 𝑌 ∧ 𝑔 Fn 𝑌) → ((𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹) ↔ 𝑓 = 𝑔))
111	97, 104, 109, 110	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → ((𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹) ↔ 𝑓 = 𝑔))
112	96, 111	imbitrid 243	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → ((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) → 𝑓 = 𝑔))
113	112	ralrimivva 3197	. 2 ⊢ (𝜑 → ∀𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))∀𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) → 𝑓 = 𝑔))
114		coeq1 5813	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
115	114	eqeq2d 2747	. . 3 ⊢ (𝑓 = 𝑔 → (𝐺 = (𝑓 ∘ 𝐹) ↔ 𝐺 = (𝑔 ∘ 𝐹)))
116	115	reu4 3689	. 2 ⊢ (∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹) ↔ (∃𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹) ∧ ∀𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))∀𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) → 𝑓 = 𝑔)))
117	95, 113, 116	sylanbrc 583	1 ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 ∃!wreu 3351 I cid 5530 ^◡ccnv 5632 ↾ cres 5635 ∘ ccom 5637 Fn wfn 6491 ⟶wf 6492 –onto→wfo 6494 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7357 qTop cqtop 17385 TopOnctopon 22259 Cn ccn 22575 Homeochmeo 23104

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-map 8767 df-qtop 17389 df-top 22243 df-topon 22260 df-cn 22578 df-hmeo 23106

This theorem is referenced by: (None)

W3C validator