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Theorem qtophmeo 23328
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 6807 . . . . . . 7 (𝐺:𝑋–ontoβ†’π‘Œ β†’ 𝐺 Fn 𝑋)
53, 4syl 17 . . . . . 6 (πœ‘ β†’ 𝐺 Fn 𝑋)
6 qtopid 23216 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐺 Fn 𝑋) β†’ 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
71, 5, 6syl2anc 584 . . . . 5 (πœ‘ β†’ 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
8 df-3an 1089 . . . . . 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 455 . . . . . 6 ((πœ‘ ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))
128, 11sylan2b 594 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))
131, 2, 7, 12qtopeu 23227 . . . 4 (πœ‘ β†’ βˆƒ!𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
14 reurex 3380 . . . 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 769 . . . 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 6807 . . . . . . . . . 10 (𝐹:𝑋–ontoβ†’π‘Œ β†’ 𝐹 Fn 𝑋)
182, 17syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐹 Fn 𝑋)
19 qtopid 23216 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
201, 18, 19syl2anc 584 . . . . . . . 8 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
21 df-3an 1089 . . . . . . . . 9 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦)) ↔ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦)))
229biimprd 247 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ ((πΊβ€˜π‘₯) = (πΊβ€˜π‘¦) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦)))
2322impr 455 . . . . . . . . 9 ((πœ‘ ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
2421, 23sylan2b 594 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
251, 3, 20, 24qtopeu 23227 . . . . . . 7 (πœ‘ β†’ βˆƒ!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔 ∘ 𝐺))
2625adantr 481 . . . . . 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 3380 . . . . . 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 23215 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹:𝑋–ontoβ†’π‘Œ) β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ))
301, 2, 29syl2anc 584 . . . . . . . . 9 (πœ‘ β†’ (𝐽 qTop 𝐹) ∈ (TopOnβ€˜π‘Œ))
3130ad2antrr 724 . . . . . . . 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 23215 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐺:𝑋–ontoβ†’π‘Œ) β†’ (𝐽 qTop 𝐺) ∈ (TopOnβ€˜π‘Œ))
331, 3, 32syl2anc 584 . . . . . . . . 9 (πœ‘ β†’ (𝐽 qTop 𝐺) ∈ (TopOnβ€˜π‘Œ))
3433ad2antrr 724 . . . . . . . 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 775 . . . . . . . 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 22760 . . . . . . . 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 1371 . . . . . . 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 769 . . . . . . . 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 22760 . . . . . . . 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 1371 . . . . . . 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 5857 . . . . . . . . 9 (β„Ž = (𝑔 ∘ 𝑓) β†’ (β„Ž ∘ 𝐹) = ((𝑔 ∘ 𝑓) ∘ 𝐹))
4241eqeq2d 2743 . . . . . . . 8 (β„Ž = (𝑔 ∘ 𝑓) β†’ (𝐹 = (β„Ž ∘ 𝐹) ↔ 𝐹 = ((𝑔 ∘ 𝑓) ∘ 𝐹)))
43 coeq1 5857 . . . . . . . . 9 (β„Ž = ( I β†Ύ π‘Œ) β†’ (β„Ž ∘ 𝐹) = (( I β†Ύ π‘Œ) ∘ 𝐹))
4443eqeq2d 2743 . . . . . . . 8 (β„Ž = ( I β†Ύ π‘Œ) β†’ (𝐹 = (β„Ž ∘ 𝐹) ↔ 𝐹 = (( I β†Ύ π‘Œ) ∘ 𝐹)))
45 simpr3 1196 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘¦))
461, 2, 20, 45qtopeu 23227 . . . . . . . . 9 (πœ‘ β†’ βˆƒ!β„Ž ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹))𝐹 = (β„Ž ∘ 𝐹))
4746ad2antrr 724 . . . . . . . 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 22777 . . . . . . . . 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 584 . . . . . . . 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 22768 . . . . . . . . . 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 724 . . . . . . . 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 771 . . . . . . . . . 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 776 . . . . . . . . . . 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 5862 . . . . . . . . . 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 2772 . . . . . . . . 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 6264 . . . . . . . . 9 ((𝑔 ∘ 𝑓) ∘ 𝐹) = (𝑔 ∘ (𝑓 ∘ 𝐹))
5856, 57eqtr4di 2790 . . . . . . . 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 6805 . . . . . . . . . . . 12 (𝐹:𝑋–ontoβ†’π‘Œ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
602, 59syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
6160ad2antrr 724 . . . . . . . . . 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 6766 . . . . . . . . . 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 2738 . . . . . . . 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 3732 . . . . . . 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 5857 . . . . . . . . 9 (β„Ž = (𝑓 ∘ 𝑔) β†’ (β„Ž ∘ 𝐺) = ((𝑓 ∘ 𝑔) ∘ 𝐺))
6766eqeq2d 2743 . . . . . . . 8 (β„Ž = (𝑓 ∘ 𝑔) β†’ (𝐺 = (β„Ž ∘ 𝐺) ↔ 𝐺 = ((𝑓 ∘ 𝑔) ∘ 𝐺)))
68 coeq1 5857 . . . . . . . . 9 (β„Ž = ( I β†Ύ π‘Œ) β†’ (β„Ž ∘ 𝐺) = (( I β†Ύ π‘Œ) ∘ 𝐺))
6968eqeq2d 2743 . . . . . . . 8 (β„Ž = ( I β†Ύ π‘Œ) β†’ (𝐺 = (β„Ž ∘ 𝐺) ↔ 𝐺 = (( I β†Ύ π‘Œ) ∘ 𝐺)))
70 simpr3 1196 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘¦))
711, 3, 7, 70qtopeu 23227 . . . . . . . . 9 (πœ‘ β†’ βˆƒ!β„Ž ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺))𝐺 = (β„Ž ∘ 𝐺))
7271ad2antrr 724 . . . . . . . 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 22777 . . . . . . . . 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 584 . . . . . . . 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 22768 . . . . . . . . . 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 724 . . . . . . . 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 5862 . . . . . . . . . 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 2772 . . . . . . . . 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 6264 . . . . . . . . 9 ((𝑓 ∘ 𝑔) ∘ 𝐺) = (𝑓 ∘ (𝑔 ∘ 𝐺))
8179, 80eqtr4di 2790 . . . . . . . 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 6805 . . . . . . . . . . . 12 (𝐺:𝑋–ontoβ†’π‘Œ β†’ 𝐺:π‘‹βŸΆπ‘Œ)
833, 82syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺:π‘‹βŸΆπ‘Œ)
8483ad2antrr 724 . . . . . . . . . 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 6766 . . . . . . . . . 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 2738 . . . . . . . 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 3732 . . . . . . 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 7295 . . . . . 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 2833 . . . . 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 3161 . . . 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 23270 . . . 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 583 . . 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 771 . . 3 ((πœ‘ ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓 ∘ 𝐹))) β†’ 𝐺 = (𝑓 ∘ 𝐹))
9515, 93, 94reximssdv 3172 . 2 (πœ‘ β†’ βˆƒπ‘“ ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
96 eqtr2 2756 . . . 4 ((𝐺 = (𝑓 ∘ 𝐹) ∧ 𝐺 = (𝑔 ∘ 𝐹)) β†’ (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
972adantr 481 . . . . 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 481 . . . . . . 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 481 . . . . . . 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 769 . . . . . . 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 23274 . . . . . . 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 1371 . . . . . 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 6834 . . . . . 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 771 . . . . . . 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 23274 . . . . . . 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 1371 . . . . . 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 6834 . . . . . 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 7292 . . . . 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 1371 . . . 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 3200 . 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 5857 . . . 4 (𝑓 = 𝑔 β†’ (𝑓 ∘ 𝐹) = (𝑔 ∘ 𝐹))
115114eqeq2d 2743 . . 3 (𝑓 = 𝑔 β†’ (𝐺 = (𝑓 ∘ 𝐹) ↔ 𝐺 = (𝑔 ∘ 𝐹)))
116115reu4 3727 . 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 583 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  βˆƒ!wreu 3374   I cid 5573  β—‘ccnv 5675   β†Ύ cres 5678   ∘ ccom 5680   Fn wfn 6538  βŸΆwf 6539  β€“ontoβ†’wfo 6541  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7411   qTop cqtop 17451  TopOnctopon 22419   Cn ccn 22735  Homeochmeo 23264
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-qtop 17455  df-top 22403  df-topon 22420  df-cn 22738  df-hmeo 23266
This theorem is referenced by: (None)
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