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Theorem qtophmeo 21430
 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 (𝜑𝐽 ∈ (TopOn‘𝑋))
qtophmeo.3 (𝜑𝐹:𝑋onto𝑌)
qtophmeo.4 (𝜑𝐺:𝑋onto𝑌)
qtophmeo.5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐺𝑥) = (𝐺𝑦)))
Assertion
Ref Expression
qtophmeo (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qtophmeo.2 . . . . 5 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 qtophmeo.3 . . . . 5 (𝜑𝐹:𝑋onto𝑌)
3 qtophmeo.4 . . . . . . 7 (𝜑𝐺:𝑋onto𝑌)
4 fofn 6030 . . . . . . 7 (𝐺:𝑋onto𝑌𝐺 Fn 𝑋)
53, 4syl 17 . . . . . 6 (𝜑𝐺 Fn 𝑋)
6 qtopid 21318 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺 Fn 𝑋) → 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
71, 5, 6syl2anc 691 . . . . 5 (𝜑𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
8 df-3an 1033 . . . . . 6 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦)))
9 qtophmeo.5 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐺𝑥) = (𝐺𝑦)))
109biimpd 218 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) → (𝐺𝑥) = (𝐺𝑦)))
1110impr 647 . . . . . 6 ((𝜑 ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐺𝑥) = (𝐺𝑦))
128, 11sylan2b 491 . . . . 5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐺𝑥) = (𝐺𝑦))
131, 2, 7, 12qtopeu 21329 . . . 4 (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))
14 reurex 3137 . . . 4 (∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓𝐹) → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))
1513, 14syl 17 . . 3 (𝜑 → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))
16 simprl 790 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)))
17 fofn 6030 . . . . . . . . . . . . 13 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
182, 17syl 17 . . . . . . . . . . . 12 (𝜑𝐹 Fn 𝑋)
19 qtopid 21318 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
201, 18, 19syl2anc 691 . . . . . . . . . . 11 (𝜑𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
21 df-3an 1033 . . . . . . . . . . . 12 ((𝑥𝑋𝑦𝑋 ∧ (𝐺𝑥) = (𝐺𝑦)) ↔ ((𝑥𝑋𝑦𝑋) ∧ (𝐺𝑥) = (𝐺𝑦)))
229biimprd 237 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐺𝑥) = (𝐺𝑦) → (𝐹𝑥) = (𝐹𝑦)))
2322impr 647 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐺𝑥) = (𝐺𝑦))) → (𝐹𝑥) = (𝐹𝑦))
2421, 23sylan2b 491 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐺𝑥) = (𝐺𝑦))) → (𝐹𝑥) = (𝐹𝑦))
251, 3, 20, 24qtopeu 21329 . . . . . . . . . 10 (𝜑 → ∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺))
2625adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → ∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺))
27 reurex 3137 . . . . . . . . 9 (∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺) → ∃𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺))
2826, 27syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → ∃𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺))
29 qtoptopon 21317 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
301, 2, 29syl2anc 691 . . . . . . . . . . . 12 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
3130ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
32 qtoptopon 21317 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺:𝑋onto𝑌) → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
331, 3, 32syl2anc 691 . . . . . . . . . . . 12 (𝜑 → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
3433ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
35 simplrl 796 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)))
36 cnf2 20863 . . . . . . . . . . 11 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))) → 𝑓:𝑌𝑌)
3731, 34, 35, 36syl3anc 1318 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝑓:𝑌𝑌)
38 simprl 790 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
39 cnf2 20863 . . . . . . . . . . 11 (((𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))) → 𝑔:𝑌𝑌)
4034, 31, 38, 39syl3anc 1318 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝑔:𝑌𝑌)
41 coeq1 5201 . . . . . . . . . . . 12 ( = (𝑔𝑓) → (𝐹) = ((𝑔𝑓) ∘ 𝐹))
4241eqeq2d 2620 . . . . . . . . . . 11 ( = (𝑔𝑓) → (𝐹 = (𝐹) ↔ 𝐹 = ((𝑔𝑓) ∘ 𝐹)))
43 coeq1 5201 . . . . . . . . . . . 12 ( = ( I ↾ 𝑌) → (𝐹) = (( I ↾ 𝑌) ∘ 𝐹))
4443eqeq2d 2620 . . . . . . . . . . 11 ( = ( I ↾ 𝑌) → (𝐹 = (𝐹) ↔ 𝐹 = (( I ↾ 𝑌) ∘ 𝐹)))
45 simpr3 1062 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
461, 2, 20, 45qtopeu 21329 . . . . . . . . . . . 12 (𝜑 → ∃! ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹))𝐹 = (𝐹))
4746ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → ∃! ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹))𝐹 = (𝐹))
48 cnco 20880 . . . . . . . . . . . 12 ((𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))) → (𝑔𝑓) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
4935, 38, 48syl2anc 691 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → (𝑔𝑓) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
50 idcn 20871 . . . . . . . . . . . . 13 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
5130, 50syl 17 . . . . . . . . . . . 12 (𝜑 → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
5251ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
53 simprr 792 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝐹 = (𝑔𝐺))
54 simplrr 797 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝐺 = (𝑓𝐹))
5554coeq2d 5206 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → (𝑔𝐺) = (𝑔 ∘ (𝑓𝐹)))
5653, 55eqtrd 2644 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝐹 = (𝑔 ∘ (𝑓𝐹)))
57 coass 5571 . . . . . . . . . . . 12 ((𝑔𝑓) ∘ 𝐹) = (𝑔 ∘ (𝑓𝐹))
5856, 57syl6eqr 2662 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝐹 = ((𝑔𝑓) ∘ 𝐹))
59 fof 6028 . . . . . . . . . . . . . . 15 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
602, 59syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:𝑋𝑌)
6160ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝐹:𝑋𝑌)
62 fcoi2 5992 . . . . . . . . . . . . 13 (𝐹:𝑋𝑌 → (( I ↾ 𝑌) ∘ 𝐹) = 𝐹)
6361, 62syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → (( I ↾ 𝑌) ∘ 𝐹) = 𝐹)
6463eqcomd 2616 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝐹 = (( I ↾ 𝑌) ∘ 𝐹))
6542, 44, 47, 49, 52, 58, 64reu2eqd 3370 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → (𝑔𝑓) = ( I ↾ 𝑌))
66 coeq1 5201 . . . . . . . . . . . 12 ( = (𝑓𝑔) → (𝐺) = ((𝑓𝑔) ∘ 𝐺))
6766eqeq2d 2620 . . . . . . . . . . 11 ( = (𝑓𝑔) → (𝐺 = (𝐺) ↔ 𝐺 = ((𝑓𝑔) ∘ 𝐺)))
68 coeq1 5201 . . . . . . . . . . . 12 ( = ( I ↾ 𝑌) → (𝐺) = (( I ↾ 𝑌) ∘ 𝐺))
6968eqeq2d 2620 . . . . . . . . . . 11 ( = ( I ↾ 𝑌) → (𝐺 = (𝐺) ↔ 𝐺 = (( I ↾ 𝑌) ∘ 𝐺)))
70 simpr3 1062 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐺𝑥) = (𝐺𝑦))) → (𝐺𝑥) = (𝐺𝑦))
711, 3, 7, 70qtopeu 21329 . . . . . . . . . . . 12 (𝜑 → ∃! ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺))𝐺 = (𝐺))
7271ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → ∃! ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺))𝐺 = (𝐺))
73 cnco 20880 . . . . . . . . . . . 12 ((𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))) → (𝑓𝑔) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
7438, 35, 73syl2anc 691 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → (𝑓𝑔) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
75 idcn 20871 . . . . . . . . . . . . 13 ((𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
7633, 75syl 17 . . . . . . . . . . . 12 (𝜑 → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
7776ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
7853coeq2d 5206 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → (𝑓𝐹) = (𝑓 ∘ (𝑔𝐺)))
7954, 78eqtrd 2644 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝐺 = (𝑓 ∘ (𝑔𝐺)))
80 coass 5571 . . . . . . . . . . . 12 ((𝑓𝑔) ∘ 𝐺) = (𝑓 ∘ (𝑔𝐺))
8179, 80syl6eqr 2662 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝐺 = ((𝑓𝑔) ∘ 𝐺))
82 fof 6028 . . . . . . . . . . . . . . 15 (𝐺:𝑋onto𝑌𝐺:𝑋𝑌)
833, 82syl 17 . . . . . . . . . . . . . 14 (𝜑𝐺:𝑋𝑌)
8483ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝐺:𝑋𝑌)
85 fcoi2 5992 . . . . . . . . . . . . 13 (𝐺:𝑋𝑌 → (( I ↾ 𝑌) ∘ 𝐺) = 𝐺)
8684, 85syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → (( I ↾ 𝑌) ∘ 𝐺) = 𝐺)
8786eqcomd 2616 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝐺 = (( I ↾ 𝑌) ∘ 𝐺))
8867, 69, 72, 74, 77, 81, 87reu2eqd 3370 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → (𝑓𝑔) = ( I ↾ 𝑌))
8937, 40, 65, 882fcoidinvd 6450 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝑓 = 𝑔)
9089, 38eqeltrd 2688 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝑓 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
9128, 90rexlimddv 3017 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
92 ishmeo 21372 . . . . . . 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))))
9316, 91, 92sylanbrc 695 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))
94 simprr 792 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝐺 = (𝑓𝐹))
9593, 94jca 553 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹)))
9695ex 449 . . . 4 (𝜑 → ((𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹)) → (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))))
9796reximdv2 2997 . . 3 (𝜑 → (∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓𝐹) → ∃𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓𝐹)))
9815, 97mpd 15 . 2 (𝜑 → ∃𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))
99 eqtr2 2630 . . . 4 ((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → (𝑓𝐹) = (𝑔𝐹))
1002adantr 480 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝐹:𝑋onto𝑌)
10130adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
10233adantr 480 . . . . . . 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 790 . . . . . . 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 21376 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))) → 𝑓:𝑌1-1-onto𝑌)
105101, 102, 103, 104syl3anc 1318 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑓:𝑌1-1-onto𝑌)
106 f1ofn 6051 . . . . . 6 (𝑓:𝑌1-1-onto𝑌𝑓 Fn 𝑌)
107105, 106syl 17 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑓 Fn 𝑌)
108 simprr 792 . . . . . . 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 21376 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))) → 𝑔:𝑌1-1-onto𝑌)
110101, 102, 108, 109syl3anc 1318 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑔:𝑌1-1-onto𝑌)
111 f1ofn 6051 . . . . . 6 (𝑔:𝑌1-1-onto𝑌𝑔 Fn 𝑌)
112110, 111syl 17 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑔 Fn 𝑌)
113 cocan2 6447 . . . . 5 ((𝐹:𝑋onto𝑌𝑓 Fn 𝑌𝑔 Fn 𝑌) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
114100, 107, 112, 113syl3anc 1318 . . . 4 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
11599, 114syl5ib 233 . . 3 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → ((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔))
116115ralrimivva 2954 . 2 (𝜑 → ∀𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))∀𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔))
117 coeq1 5201 . . . 4 (𝑓 = 𝑔 → (𝑓𝐹) = (𝑔𝐹))
118117eqeq2d 2620 . . 3 (𝑓 = 𝑔 → (𝐺 = (𝑓𝐹) ↔ 𝐺 = (𝑔𝐹)))
119118reu4 3367 . 2 (∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓𝐹) ↔ (∃𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓𝐹) ∧ ∀𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))∀𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔)))
12098, 116, 119sylanbrc 695 1 (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   I cid 4948  ◡ccnv 5037   ↾ cres 5040   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   qTop cqtop 15986  TopOnctopon 20518   Cn ccn 20838  Homeochmeo 21366 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-qtop 15990  df-top 20521  df-topon 20523  df-cn 20841  df-hmeo 21368 This theorem is referenced by: (None)
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