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Theorem qtophmeo 21620

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 6117	. . . . . . 7 ⊢ (𝐺:𝑋–onto→𝑌 → 𝐺 Fn 𝑋)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝑋)
6		qtopid 21508	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺 Fn 𝑋) → 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
7	1, 5, 6	syl2anc 693	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
8		df-3an 1039	. . . . . 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 219	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐺‘𝑥) = (𝐺‘𝑦)))
11	10	impr 649	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))
12	8, 11	sylan2b 492	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))
13	1, 2, 7, 12	qtopeu 21519	. . . 4 ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
14		reurex 3160	. . . 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 794	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)))
17		fofn 6117	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
18	2, 17	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn 𝑋)
19		qtopid 21508	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
20	1, 18, 19	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
21		df-3an 1039	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐺‘𝑥) = (𝐺‘𝑦)) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐺‘𝑥) = (𝐺‘𝑦)))
22	9	biimprd 238	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐺‘𝑥) = (𝐺‘𝑦) → (𝐹‘𝑥) = (𝐹‘𝑦)))
23	22	impr 649	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐺‘𝑥) = (𝐺‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
24	21, 23	sylan2b 492	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐺‘𝑥) = (𝐺‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
25	1, 3, 20, 24	qtopeu 21519	. . . . . . . . . 10 ⊢ (𝜑 → ∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺))
26	25	adantr 481	. . . . . . . . 9 ⊢ ((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 3160	. . . . . . . . 9 ⊢ (∃!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	. . . . . . . 8 ⊢ ((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 21507	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
30	1, 2, 29	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
31	30	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((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 21507	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺:𝑋–onto→𝑌) → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
33	1, 3, 32	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
34	33	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((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 800	. . . . . . . . . . 11 ⊢ (((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 21053	. . . . . . . . . . 11 ⊢ (((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 1326	. . . . . . . . . 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:𝑌⟶𝑌)
38		simprl 794	. . . . . . . . . . 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 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
39		cnf2 21053	. . . . . . . . . . 11 ⊢ (((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 1326	. . . . . . . . . 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:𝑌⟶𝑌)
41		coeq1 5279	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑔 ∘ 𝑓) → (ℎ ∘ 𝐹) = ((𝑔 ∘ 𝑓) ∘ 𝐹))
42	41	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (ℎ = (𝑔 ∘ 𝑓) → (𝐹 = (ℎ ∘ 𝐹) ↔ 𝐹 = ((𝑔 ∘ 𝑓) ∘ 𝐹)))
43		coeq1 5279	. . . . . . . . . . . 12 ⊢ (ℎ = ( I ↾ 𝑌) → (ℎ ∘ 𝐹) = (( I ↾ 𝑌) ∘ 𝐹))
44	43	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (ℎ = ( I ↾ 𝑌) → (𝐹 = (ℎ ∘ 𝐹) ↔ 𝐹 = (( I ↾ 𝑌) ∘ 𝐹)))
45		simpr3 1069	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
46	1, 2, 20, 45	qtopeu 21519	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃!ℎ ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹))𝐹 = (ℎ ∘ 𝐹))
47	46	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((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 21070	. . . . . . . . . . . 12 ⊢ ((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 693	. . . . . . . . . . 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) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
50		idcn 21061	. . . . . . . . . . . . 13 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
51	30, 50	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
52	51	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((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 796	. . . . . . . . . . . . 13 ⊢ (((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 801	. . . . . . . . . . . . . 14 ⊢ (((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 5284	. . . . . . . . . . . . 13 ⊢ (((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 2656	. . . . . . . . . . . 12 ⊢ (((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 5654	. . . . . . . . . . . 12 ⊢ ((𝑔 ∘ 𝑓) ∘ 𝐹) = (𝑔 ∘ (𝑓 ∘ 𝐹))
58	56, 57	syl6eqr 2674	. . . . . . . . . . 11 ⊢ (((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 6115	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
60	2, 59	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
61	60	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((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 6079	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑋⟶𝑌 → (( I ↾ 𝑌) ∘ 𝐹) = 𝐹)
63	61, 62	syl 17	. . . . . . . . . . . 12 ⊢ (((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 2628	. . . . . . . . . . 11 ⊢ (((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 3403	. . . . . . . . . 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 ∘ 𝑓) = ( I ↾ 𝑌))
66		coeq1 5279	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ ∘ 𝐺) = ((𝑓 ∘ 𝑔) ∘ 𝐺))
67	66	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (𝐺 = (ℎ ∘ 𝐺) ↔ 𝐺 = ((𝑓 ∘ 𝑔) ∘ 𝐺)))
68		coeq1 5279	. . . . . . . . . . . 12 ⊢ (ℎ = ( I ↾ 𝑌) → (ℎ ∘ 𝐺) = (( I ↾ 𝑌) ∘ 𝐺))
69	68	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (ℎ = ( I ↾ 𝑌) → (𝐺 = (ℎ ∘ 𝐺) ↔ 𝐺 = (( I ↾ 𝑌) ∘ 𝐺)))
70		simpr3 1069	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐺‘𝑥) = (𝐺‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))
71	1, 3, 7, 70	qtopeu 21519	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃!ℎ ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺))𝐺 = (ℎ ∘ 𝐺))
72	71	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((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 21070	. . . . . . . . . . . 12 ⊢ ((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 693	. . . . . . . . . . 11 ⊢ (((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 21061	. . . . . . . . . . . . 13 ⊢ ((𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
76	33, 75	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
77	76	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((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 5284	. . . . . . . . . . . . 13 ⊢ (((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 2656	. . . . . . . . . . . 12 ⊢ (((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 5654	. . . . . . . . . . . 12 ⊢ ((𝑓 ∘ 𝑔) ∘ 𝐺) = (𝑓 ∘ (𝑔 ∘ 𝐺))
81	79, 80	syl6eqr 2674	. . . . . . . . . . 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 ∘ 𝑔) ∘ 𝐺))
82		fof 6115	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝑋–onto→𝑌 → 𝐺:𝑋⟶𝑌)
83	3, 82	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:𝑋⟶𝑌)
84	83	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((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 6079	. . . . . . . . . . . . 13 ⊢ (𝐺:𝑋⟶𝑌 → (( I ↾ 𝑌) ∘ 𝐺) = 𝐺)
86	84, 85	syl 17	. . . . . . . . . . . 12 ⊢ (((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 2628	. . . . . . . . . . 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 = (( I ↾ 𝑌) ∘ 𝐺))
88	67, 69, 72, 74, 77, 81, 87	reu2eqd 3403	. . . . . . . . . 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) = ( I ↾ 𝑌))
89	37, 40, 65, 88	2fcoidinvd 6550	. . . . . . . . 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)
90	89, 38	eqeltrd 2701	. . . . . . . 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 𝐺) Cn (𝐽 qTop 𝐹)))
91	28, 90	rexlimddv 3035	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → ^◡𝑓 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
92		ishmeo 21562	. . . . . . 7 ⊢ (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 698	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))
94		simprr 796	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → 𝐺 = (𝑓 ∘ 𝐹))
95	93, 94	jca 554	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹)))
96	95	ex 450	. . . 4 ⊢ (𝜑 → ((𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹)) → (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))))
97	96	reximdv2 3014	. . 3 ⊢ (𝜑 → (∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹) → ∃𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹)))
98	15, 97	mpd 15	. 2 ⊢ (𝜑 → ∃𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
99		eqtr2 2642	. . . 4 ⊢ ((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) → (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
100	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)
101	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))
102	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))
103		simprl 794	. . . . . . 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)))
104		hmeof1o2 21566	. . . . . . 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)
105	101, 102, 103, 104	syl3anc 1326	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑓:𝑌–1-1-onto→𝑌)
106		f1ofn 6138	. . . . . 6 ⊢ (𝑓:𝑌–1-1-onto→𝑌 → 𝑓 Fn 𝑌)
107	105, 106	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑓 Fn 𝑌)
108		simprr 796	. . . . . . 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)))
109		hmeof1o2 21566	. . . . . . 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)
110	101, 102, 108, 109	syl3anc 1326	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑔:𝑌–1-1-onto→𝑌)
111		f1ofn 6138	. . . . . 6 ⊢ (𝑔:𝑌–1-1-onto→𝑌 → 𝑔 Fn 𝑌)
112	110, 111	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑔 Fn 𝑌)
113		cocan2 6547	. . . . 5 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑓 Fn 𝑌 ∧ 𝑔 Fn 𝑌) → ((𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹) ↔ 𝑓 = 𝑔))
114	100, 107, 112, 113	syl3anc 1326	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → ((𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹) ↔ 𝑓 = 𝑔))
115	99, 114	syl5ib 234	. . 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))
116	115	ralrimivva 2971	. 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))
117		coeq1 5279	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
118	117	eqeq2d 2632	. . 3 ⊢ (𝑓 = 𝑔 → (𝐺 = (𝑓 ∘ 𝐹) ↔ 𝐺 = (𝑔 ∘ 𝐹)))
119	118	reu4 3400	. 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)))
120	98, 116, 119	sylanbrc 698	1 ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ∃!wreu 2914 I cid 5023 ^◡ccnv 5113 ↾ cres 5116 ∘ ccom 5118 Fn wfn 5883 ⟶wf 5884 –onto→wfo 5886 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 qTop cqtop 16163 TopOnctopon 20715 Cn ccn 21028 Homeochmeo 21556

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-qtop 16167 df-top 20699 df-topon 20716 df-cn 21031 df-hmeo 21558

This theorem is referenced by: (None)

W3C validator