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Theorem qtophmeo 23184
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 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
qtophmeo.3 (πœ‘ β†’ 𝐹:𝑋–ontoβ†’π‘Œ)
qtophmeo.4 (πœ‘ β†’ 𝐺:𝑋–ontoβ†’π‘Œ)
qtophmeo.5 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ↔ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦)))
Assertion
Ref Expression
qtophmeo (πœ‘ β†’ βˆƒ!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
Distinct variable groups:   π‘₯,𝑓,𝑦,𝐹   𝑓,𝐺,π‘₯,𝑦   𝑓,𝐽,π‘₯,𝑦   πœ‘,𝑓,π‘₯,𝑦   π‘₯,𝑋,𝑦   𝑓,π‘Œ,π‘₯
Allowed substitution hints:   𝑋(𝑓)   π‘Œ(𝑦)

Proof of Theorem qtophmeo
Dummy variables 𝑔 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qtophmeo.2 . . . . 5 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 qtophmeo.3 . . . . 5 (πœ‘ β†’ 𝐹:𝑋–ontoβ†’π‘Œ)
3 qtophmeo.4 . . . . . . 7 (πœ‘ β†’ 𝐺:𝑋–ontoβ†’π‘Œ)
4 fofn 6759 . . . . . . 7 (𝐺:𝑋–ontoβ†’π‘Œ β†’ 𝐺 Fn 𝑋)
53, 4syl 17 . . . . . 6 (πœ‘ β†’ 𝐺 Fn 𝑋)
6 qtopid 23072 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐺 Fn 𝑋) β†’ 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
71, 5, 6syl2anc 585 . . . . 5 (πœ‘ β†’ 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
8 df-3an 1090 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦)) ↔ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦)))
9 qtophmeo.5 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ↔ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦)))
109biimpd 228 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦)))
1110impr 456 . . . . . 6 ((πœ‘ ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))
128, 11sylan2b 595 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))
131, 2, 7, 12qtopeu 23083 . . . 4 (πœ‘ β†’ βˆƒ!𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
14 reurex 3356 . . . 4 (βˆƒ!𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹) β†’ βˆƒπ‘“ ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
1513, 14syl 17 . . 3 (πœ‘ β†’ βˆƒπ‘“ ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
16 simprl 770 . . . 4 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) β†’ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)))
17 fofn 6759 . . . . . . . . . 10 (𝐹:𝑋–ontoβ†’π‘Œ β†’ 𝐹 Fn 𝑋)
182, 17syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐹 Fn 𝑋)
19 qtopid 23072 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
201, 18, 19syl2anc 585 . . . . . . . 8 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
21 df-3an 1090 . . . . . . . . 9 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦)) ↔ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦)))
229biimprd 248 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ ((πΊβ€˜π‘₯) = (πΊβ€˜π‘¦) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦)))
2322impr 456 . . . . . . . . 9 ((πœ‘ ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
2421, 23sylan2b 595 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
251, 3, 20, 24qtopeu 23083 . . . . . . 7 (πœ‘ β†’ βˆƒ!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺))
2625adantr 482 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) β†’ βˆƒ!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺))
27 reurex 3356 . . . . . 6 (βˆƒ!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺) β†’ βˆƒπ‘” ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺))
2826, 27syl 17 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) β†’ βˆƒπ‘” ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺))
29 qtoptopon 23071 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ))
301, 2, 29syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ))
3130ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ))
32 qtoptopon 23071 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐺:𝑋–ontoβ†’π‘Œ) β†’ (𝐽 qTop 𝐺) ∈ (TopOnβ€˜π‘Œ))
331, 3, 32syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ (𝐽 qTop 𝐺) ∈ (TopOnβ€˜π‘Œ))
3433ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ (𝐽 qTop 𝐺) ∈ (TopOnβ€˜π‘Œ))
35 simplrl 776 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)))
36 cnf2 22616 . . . . . . . 8 (((𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ) ∧ (𝐽 qTop 𝐺) ∈ (TopOnβ€˜π‘Œ) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))) β†’ 𝑓:π‘ŒβŸΆπ‘Œ)
3731, 34, 35, 36syl3anc 1372 . . . . . . 7 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝑓:π‘ŒβŸΆπ‘Œ)
38 simprl 770 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
39 cnf2 22616 . . . . . . . 8 (((𝐽 qTop 𝐺) ∈ (TopOnβ€˜π‘Œ) ∧ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ) ∧ 𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))) β†’ 𝑔:π‘ŒβŸΆπ‘Œ)
4034, 31, 38, 39syl3anc 1372 . . . . . . 7 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝑔:π‘ŒβŸΆπ‘Œ)
41 coeq1 5814 . . . . . . . . 9 (β„Ž = (𝑔 ∘ 𝑓) β†’ (β„Ž ∘ 𝐹) = ((𝑔 ∘ 𝑓) ∘ 𝐹))
4241eqeq2d 2744 . . . . . . . 8 (β„Ž = (𝑔 ∘ 𝑓) β†’ (𝐹 = (β„Ž ∘ 𝐹) ↔ 𝐹 = ((𝑔 ∘ 𝑓) ∘ 𝐹)))
43 coeq1 5814 . . . . . . . . 9 (β„Ž = ( I β†Ύ π‘Œ) β†’ (β„Ž ∘ 𝐹) = (( I β†Ύ π‘Œ) ∘ 𝐹))
4443eqeq2d 2744 . . . . . . . 8 (β„Ž = ( I β†Ύ π‘Œ) β†’ (𝐹 = (β„Ž ∘ 𝐹) ↔ 𝐹 = (( I β†Ύ π‘Œ) ∘ 𝐹)))
45 simpr3 1197 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
461, 2, 20, 45qtopeu 23083 . . . . . . . . 9 (πœ‘ β†’ βˆƒ!β„Ž ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹))𝐹 = (β„Ž ∘ 𝐹))
4746ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ βˆƒ!β„Ž ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹))𝐹 = (β„Ž ∘ 𝐹))
48 cnco 22633 . . . . . . . . 9 ((𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))) β†’ (𝑔 ∘ 𝑓) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
4935, 38, 48syl2anc 585 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ (𝑔 ∘ 𝑓) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
50 idcn 22624 . . . . . . . . . 10 ((𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ) β†’ ( I β†Ύ π‘Œ) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
5130, 50syl 17 . . . . . . . . 9 (πœ‘ β†’ ( I β†Ύ π‘Œ) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
5251ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ ( I β†Ύ π‘Œ) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
53 simprr 772 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝐹 = (𝑔 ∘ 𝐺))
54 simplrr 777 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝐺 = (𝑓 ∘ 𝐹))
5554coeq2d 5819 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ (𝑔 ∘ 𝐺) = (𝑔 ∘ (𝑓 ∘ 𝐹)))
5653, 55eqtrd 2773 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝐹 = (𝑔 ∘ (𝑓 ∘ 𝐹)))
57 coass 6218 . . . . . . . . 9 ((𝑔 ∘ 𝑓) ∘ 𝐹) = (𝑔 ∘ (𝑓 ∘ 𝐹))
5856, 57eqtr4di 2791 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝐹 = ((𝑔 ∘ 𝑓) ∘ 𝐹))
59 fof 6757 . . . . . . . . . . . 12 (𝐹:𝑋–ontoβ†’π‘Œ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
602, 59syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
6160ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
62 fcoi2 6718 . . . . . . . . . 10 (𝐹:π‘‹βŸΆπ‘Œ β†’ (( I β†Ύ π‘Œ) ∘ 𝐹) = 𝐹)
6361, 62syl 17 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ (( I β†Ύ π‘Œ) ∘ 𝐹) = 𝐹)
6463eqcomd 2739 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝐹 = (( I β†Ύ π‘Œ) ∘ 𝐹))
6542, 44, 47, 49, 52, 58, 64reu2eqd 3695 . . . . . . 7 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ (𝑔 ∘ 𝑓) = ( I β†Ύ π‘Œ))
66 coeq1 5814 . . . . . . . . 9 (β„Ž = (𝑓 ∘ 𝑔) β†’ (β„Ž ∘ 𝐺) = ((𝑓 ∘ 𝑔) ∘ 𝐺))
6766eqeq2d 2744 . . . . . . . 8 (β„Ž = (𝑓 ∘ 𝑔) β†’ (𝐺 = (β„Ž ∘ 𝐺) ↔ 𝐺 = ((𝑓 ∘ 𝑔) ∘ 𝐺)))
68 coeq1 5814 . . . . . . . . 9 (β„Ž = ( I β†Ύ π‘Œ) β†’ (β„Ž ∘ 𝐺) = (( I β†Ύ π‘Œ) ∘ 𝐺))
6968eqeq2d 2744 . . . . . . . 8 (β„Ž = ( I β†Ύ π‘Œ) β†’ (𝐺 = (β„Ž ∘ 𝐺) ↔ 𝐺 = (( I β†Ύ π‘Œ) ∘ 𝐺)))
70 simpr3 1197 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))
711, 3, 7, 70qtopeu 23083 . . . . . . . . 9 (πœ‘ β†’ βˆƒ!β„Ž ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺))𝐺 = (β„Ž ∘ 𝐺))
7271ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ βˆƒ!β„Ž ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺))𝐺 = (β„Ž ∘ 𝐺))
73 cnco 22633 . . . . . . . . 9 ((𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))) β†’ (𝑓 ∘ 𝑔) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
7438, 35, 73syl2anc 585 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ (𝑓 ∘ 𝑔) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
75 idcn 22624 . . . . . . . . . 10 ((𝐽 qTop 𝐺) ∈ (TopOnβ€˜π‘Œ) β†’ ( I β†Ύ π‘Œ) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
7633, 75syl 17 . . . . . . . . 9 (πœ‘ β†’ ( I β†Ύ π‘Œ) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
7776ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ ( I β†Ύ π‘Œ) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
7853coeq2d 5819 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ (𝑓 ∘ 𝐹) = (𝑓 ∘ (𝑔 ∘ 𝐺)))
7954, 78eqtrd 2773 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝐺 = (𝑓 ∘ (𝑔 ∘ 𝐺)))
80 coass 6218 . . . . . . . . 9 ((𝑓 ∘ 𝑔) ∘ 𝐺) = (𝑓 ∘ (𝑔 ∘ 𝐺))
8179, 80eqtr4di 2791 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝐺 = ((𝑓 ∘ 𝑔) ∘ 𝐺))
82 fof 6757 . . . . . . . . . . . 12 (𝐺:𝑋–ontoβ†’π‘Œ β†’ 𝐺:π‘‹βŸΆπ‘Œ)
833, 82syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺:π‘‹βŸΆπ‘Œ)
8483ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝐺:π‘‹βŸΆπ‘Œ)
85 fcoi2 6718 . . . . . . . . . 10 (𝐺:π‘‹βŸΆπ‘Œ β†’ (( I β†Ύ π‘Œ) ∘ 𝐺) = 𝐺)
8684, 85syl 17 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ (( I β†Ύ π‘Œ) ∘ 𝐺) = 𝐺)
8786eqcomd 2739 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ 𝐺 = (( I β†Ύ π‘Œ) ∘ 𝐺))
8867, 69, 72, 74, 77, 81, 87reu2eqd 3695 . . . . . . 7 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ (𝑓 ∘ 𝑔) = ( I β†Ύ π‘Œ))
8937, 40, 65, 882fcoidinvd 7242 . . . . . 6 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ ◑𝑓 = 𝑔)
9089, 38eqeltrd 2834 . . . . 5 (((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔 ∘ 𝐺))) β†’ ◑𝑓 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
9128, 90rexlimddv 3155 . . . 4 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) β†’ ◑𝑓 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
92 ishmeo 23126 . . . 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))))
9316, 91, 92sylanbrc 584 . . 3 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) β†’ 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))
94 simprr 772 . . 3 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) β†’ 𝐺 = (𝑓 ∘ 𝐹))
9515, 93, 94reximssdv 3166 . 2 (πœ‘ β†’ βˆƒπ‘“ ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
96 eqtr2 2757 . . . 4 ((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) β†’ (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
972adantr 482 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) β†’ 𝐹:𝑋–ontoβ†’π‘Œ)
9830adantr 482 . . . . . . 7 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ))
9933adantr 482 . . . . . . 7 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) β†’ (𝐽 qTop 𝐺) ∈ (TopOnβ€˜π‘Œ))
100 simprl 770 . . . . . . 7 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) β†’ 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))
101 hmeof1o2 23130 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ) ∧ (𝐽 qTop 𝐺) ∈ (TopOnβ€˜π‘Œ) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))) β†’ 𝑓:π‘Œβ€“1-1-ontoβ†’π‘Œ)
10298, 99, 100, 101syl3anc 1372 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) β†’ 𝑓:π‘Œβ€“1-1-ontoβ†’π‘Œ)
103 f1ofn 6786 . . . . . 6 (𝑓:π‘Œβ€“1-1-ontoβ†’π‘Œ β†’ 𝑓 Fn π‘Œ)
104102, 103syl 17 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) β†’ 𝑓 Fn π‘Œ)
105 simprr 772 . . . . . . 7 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) β†’ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))
106 hmeof1o2 23130 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ) ∧ (𝐽 qTop 𝐺) ∈ (TopOnβ€˜π‘Œ) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))) β†’ 𝑔:π‘Œβ€“1-1-ontoβ†’π‘Œ)
10798, 99, 105, 106syl3anc 1372 . . . . . 6 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) β†’ 𝑔:π‘Œβ€“1-1-ontoβ†’π‘Œ)
108 f1ofn 6786 . . . . . 6 (𝑔:π‘Œβ€“1-1-ontoβ†’π‘Œ β†’ 𝑔 Fn π‘Œ)
109107, 108syl 17 . . . . 5 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) β†’ 𝑔 Fn π‘Œ)
110 cocan2 7239 . . . . 5 ((𝐹:𝑋–ontoβ†’π‘Œ ∧ 𝑓 Fn π‘Œ ∧ 𝑔 Fn π‘Œ) β†’ ((𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹) ↔ 𝑓 = 𝑔))
11197, 104, 109, 110syl3anc 1372 . . . 4 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) β†’ ((𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹) ↔ 𝑓 = 𝑔))
11296, 111imbitrid 243 . . 3 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) β†’ ((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) β†’ 𝑓 = 𝑔))
113112ralrimivva 3194 . 2 (πœ‘ β†’ βˆ€π‘“ ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))βˆ€π‘” ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) β†’ 𝑓 = 𝑔))
114 coeq1 5814 . . . 4 (𝑓 = 𝑔 β†’ (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
115114eqeq2d 2744 . . 3 (𝑓 = 𝑔 β†’ (𝐺 = (𝑓 ∘ 𝐹) ↔ 𝐺 = (𝑔 ∘ 𝐹)))
116115reu4 3690 . 2 (βˆƒ!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹) ↔ (βˆƒπ‘“ ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹) ∧ βˆ€π‘“ ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))βˆ€π‘” ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) β†’ 𝑓 = 𝑔)))
11795, 113, 116sylanbrc 584 1 (πœ‘ β†’ βˆƒ!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  βˆƒ!wreu 3350   I cid 5531  β—‘ccnv 5633   β†Ύ cres 5636   ∘ ccom 5638   Fn wfn 6492  βŸΆwf 6493  β€“ontoβ†’wfo 6495  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358   qTop cqtop 17390  TopOnctopon 22275   Cn ccn 22591  Homeochmeo 23120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-qtop 17394  df-top 22259  df-topon 22276  df-cn 22594  df-hmeo 23122
This theorem is referenced by: (None)
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