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Theorem qtophmeo 23773

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 6756	. . . . . . 7 ⊢ (𝐺:𝑋–onto→𝑌 → 𝐺 Fn 𝑋)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝑋)
6		qtopid 23661	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺 Fn 𝑋) → 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
7	1, 5, 6	syl2anc 585	. . . . 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 229	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐺‘𝑥) = (𝐺‘𝑦)))
11	10	impr 454	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))
12	8, 11	sylan2b 595	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))
13	1, 2, 7, 12	qtopeu 23672	. . . 4 ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
14		reurex 3356	. . . 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 771	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)))
17		fofn 6756	. . . . . . . . . 10 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹 Fn 𝑋)
18	2, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝑋)
19		qtopid 23661	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
20	1, 18, 19	syl2anc 585	. . . . . . . 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 248	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐺‘𝑥) = (𝐺‘𝑦) → (𝐹‘𝑥) = (𝐹‘𝑦)))
23	22	impr 454	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐺‘𝑥) = (𝐺‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
24	21, 23	sylan2b 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐺‘𝑥) = (𝐺‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
25	1, 3, 20, 24	qtopeu 23672	. . . . . . 7 ⊢ (𝜑 → ∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺))
26	25	adantr 480	. . . . . 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 3356	. . . . . 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 23660	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
30	1, 2, 29	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
31	30	ad2antrr 727	. . . . . . . 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 23660	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺:𝑋–onto→𝑌) → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
33	1, 3, 32	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
34	33	ad2antrr 727	. . . . . . . 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 777	. . . . . . . 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 23205	. . . . . . . 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 1374	. . . . . . 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 771	. . . . . . . 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 23205	. . . . . . . 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 1374	. . . . . . 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 5814	. . . . . . . . 9 ⊢ (ℎ = (𝑔 ∘ 𝑓) → (ℎ ∘ 𝐹) = ((𝑔 ∘ 𝑓) ∘ 𝐹))
42	41	eqeq2d 2748	. . . . . . . 8 ⊢ (ℎ = (𝑔 ∘ 𝑓) → (𝐹 = (ℎ ∘ 𝐹) ↔ 𝐹 = ((𝑔 ∘ 𝑓) ∘ 𝐹)))
43		coeq1 5814	. . . . . . . . 9 ⊢ (ℎ = ( I ↾ 𝑌) → (ℎ ∘ 𝐹) = (( I ↾ 𝑌) ∘ 𝐹))
44	43	eqeq2d 2748	. . . . . . . 8 ⊢ (ℎ = ( I ↾ 𝑌) → (𝐹 = (ℎ ∘ 𝐹) ↔ 𝐹 = (( I ↾ 𝑌) ∘ 𝐹)))
45		simpr3 1198	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
46	1, 2, 20, 45	qtopeu 23672	. . . . . . . . 9 ⊢ (𝜑 → ∃!ℎ ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹))𝐹 = (ℎ ∘ 𝐹))
47	46	ad2antrr 727	. . . . . . . 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 23222	. . . . . . . . 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 585	. . . . . . . 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 23213	. . . . . . . . . 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 727	. . . . . . . 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 773	. . . . . . . . . 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 778	. . . . . . . . . . 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 5819	. . . . . . . . . 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 2772	. . . . . . . . 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 6232	. . . . . . . . 9 ⊢ ((𝑔 ∘ 𝑓) ∘ 𝐹) = (𝑔 ∘ (𝑓 ∘ 𝐹))
58	56, 57	eqtr4di 2790	. . . . . . . 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 6754	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
60	2, 59	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
61	60	ad2antrr 727	. . . . . . . . . 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 2743	. . . . . . . 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 3696	. . . . . . 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 5814	. . . . . . . . 9 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ ∘ 𝐺) = ((𝑓 ∘ 𝑔) ∘ 𝐺))
67	66	eqeq2d 2748	. . . . . . . 8 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (𝐺 = (ℎ ∘ 𝐺) ↔ 𝐺 = ((𝑓 ∘ 𝑔) ∘ 𝐺)))
68		coeq1 5814	. . . . . . . . 9 ⊢ (ℎ = ( I ↾ 𝑌) → (ℎ ∘ 𝐺) = (( I ↾ 𝑌) ∘ 𝐺))
69	68	eqeq2d 2748	. . . . . . . 8 ⊢ (ℎ = ( I ↾ 𝑌) → (𝐺 = (ℎ ∘ 𝐺) ↔ 𝐺 = (( I ↾ 𝑌) ∘ 𝐺)))
70		simpr3 1198	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝐺‘𝑥) = (𝐺‘𝑦))) → (𝐺‘𝑥) = (𝐺‘𝑦))
71	1, 3, 7, 70	qtopeu 23672	. . . . . . . . 9 ⊢ (𝜑 → ∃!ℎ ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺))𝐺 = (ℎ ∘ 𝐺))
72	71	ad2antrr 727	. . . . . . . 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 23222	. . . . . . . . 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 585	. . . . . . . 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 23213	. . . . . . . . . 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 727	. . . . . . . 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 5819	. . . . . . . . . 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 2772	. . . . . . . . 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 6232	. . . . . . . . 9 ⊢ ((𝑓 ∘ 𝑔) ∘ 𝐺) = (𝑓 ∘ (𝑔 ∘ 𝐺))
81	79, 80	eqtr4di 2790	. . . . . . . 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 6754	. . . . . . . . . . . 12 ⊢ (𝐺:𝑋–onto→𝑌 → 𝐺:𝑋⟶𝑌)
83	3, 82	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:𝑋⟶𝑌)
84	83	ad2antrr 727	. . . . . . . . . 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 2743	. . . . . . . 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 3696	. . . . . . 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 7251	. . . . . 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 2837	. . . . 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 3145	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → ^◡𝑓 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
92		ishmeo 23715	. . . 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 584	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))
94		simprr 773	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) → 𝐺 = (𝑓 ∘ 𝐹))
95	15, 93, 94	reximssdv 3156	. 2 ⊢ (𝜑 → ∃𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
96		eqtr2 2758	. . . 4 ⊢ ((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) → (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
97	2	adantr 480	. . . . 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 480	. . . . . . 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 480	. . . . . . 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 771	. . . . . . 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 23719	. . . . . . 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 1374	. . . . . 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 6783	. . . . . 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 773	. . . . . . 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 23719	. . . . . . 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 1374	. . . . . 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 6783	. . . . . 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 7248	. . . . 5 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑓 Fn 𝑌 ∧ 𝑔 Fn 𝑌) → ((𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹) ↔ 𝑓 = 𝑔))
111	97, 104, 109, 110	syl3anc 1374	. . . 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 244	. . 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 3181	. 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 5814	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
115	114	eqeq2d 2748	. . 3 ⊢ (𝑓 = 𝑔 → (𝐺 = (𝑓 ∘ 𝐹) ↔ 𝐺 = (𝑔 ∘ 𝐹)))
116	115	reu4 3691	. 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 584	1 ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ∃!wreu 3350 I cid 5526 ^◡ccnv 5631 ↾ cres 5634 ∘ ccom 5636 Fn wfn 6495 ⟶wf 6496 –onto→wfo 6498 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 qTop cqtop 17436 TopOnctopon 22866 Cn ccn 23180 Homeochmeo 23709

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-map 8777 df-qtop 17440 df-top 22850 df-topon 22867 df-cn 23183 df-hmeo 23711

This theorem is referenced by: (None)

W3C validator