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Theorem qtophmeo 21841
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 6258 . . . . . . 7 (𝐺:𝑋onto𝑌𝐺 Fn 𝑋)
53, 4syl 17 . . . . . 6 (𝜑𝐺 Fn 𝑋)
6 qtopid 21729 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺 Fn 𝑋) → 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
71, 5, 6syl2anc 573 . . . . 5 (𝜑𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
8 df-3an 1073 . . . . . 6 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦)))
9 qtophmeo.5 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐺𝑥) = (𝐺𝑦)))
109biimpd 219 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) → (𝐺𝑥) = (𝐺𝑦)))
1110impr 442 . . . . . 6 ((𝜑 ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐺𝑥) = (𝐺𝑦))
128, 11sylan2b 581 . . . . 5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐺𝑥) = (𝐺𝑦))
131, 2, 7, 12qtopeu 21740 . . . 4 (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))
14 reurex 3309 . . . 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 754 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)))
17 fofn 6258 . . . . . . . . . . . . 13 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
182, 17syl 17 . . . . . . . . . . . 12 (𝜑𝐹 Fn 𝑋)
19 qtopid 21729 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
201, 18, 19syl2anc 573 . . . . . . . . . . 11 (𝜑𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
21 df-3an 1073 . . . . . . . . . . . 12 ((𝑥𝑋𝑦𝑋 ∧ (𝐺𝑥) = (𝐺𝑦)) ↔ ((𝑥𝑋𝑦𝑋) ∧ (𝐺𝑥) = (𝐺𝑦)))
229biimprd 238 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐺𝑥) = (𝐺𝑦) → (𝐹𝑥) = (𝐹𝑦)))
2322impr 442 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐺𝑥) = (𝐺𝑦))) → (𝐹𝑥) = (𝐹𝑦))
2421, 23sylan2b 581 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐺𝑥) = (𝐺𝑦))) → (𝐹𝑥) = (𝐹𝑦))
251, 3, 20, 24qtopeu 21740 . . . . . . . . . 10 (𝜑 → ∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺))
2625adantr 466 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → ∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺))
27 reurex 3309 . . . . . . . . 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 21728 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
301, 2, 29syl2anc 573 . . . . . . . . . . . 12 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
3130ad2antrr 705 . . . . . . . . . . 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 21728 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺:𝑋onto𝑌) → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
331, 3, 32syl2anc 573 . . . . . . . . . . . 12 (𝜑 → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
3433ad2antrr 705 . . . . . . . . . . 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 762 . . . . . . . . . . 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 21274 . . . . . . . . . . 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 1476 . . . . . . . . . 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 754 . . . . . . . . . . 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 21274 . . . . . . . . . . 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 1476 . . . . . . . . . 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 5418 . . . . . . . . . . . 12 ( = (𝑔𝑓) → (𝐹) = ((𝑔𝑓) ∘ 𝐹))
4241eqeq2d 2781 . . . . . . . . . . 11 ( = (𝑔𝑓) → (𝐹 = (𝐹) ↔ 𝐹 = ((𝑔𝑓) ∘ 𝐹)))
43 coeq1 5418 . . . . . . . . . . . 12 ( = ( I ↾ 𝑌) → (𝐹) = (( I ↾ 𝑌) ∘ 𝐹))
4443eqeq2d 2781 . . . . . . . . . . 11 ( = ( I ↾ 𝑌) → (𝐹 = (𝐹) ↔ 𝐹 = (( I ↾ 𝑌) ∘ 𝐹)))
45 simpr3 1237 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
461, 2, 20, 45qtopeu 21740 . . . . . . . . . . . 12 (𝜑 → ∃! ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹))𝐹 = (𝐹))
4746ad2antrr 705 . . . . . . . . . . 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 21291 . . . . . . . . . . . 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 573 . . . . . . . . . . 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 21282 . . . . . . . . . . . . 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 705 . . . . . . . . . . 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 756 . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . 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 5423 . . . . . . . . . . . . 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 2805 . . . . . . . . . . . 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 5798 . . . . . . . . . . . 12 ((𝑔𝑓) ∘ 𝐹) = (𝑔 ∘ (𝑓𝐹))
5856, 57syl6eqr 2823 . . . . . . . . . . 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 6256 . . . . . . . . . . . . . . 15 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
602, 59syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:𝑋𝑌)
6160ad2antrr 705 . . . . . . . . . . . . 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 6219 . . . . . . . . . . . . 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 2777 . . . . . . . . . . 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 3555 . . . . . . . . . 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 5418 . . . . . . . . . . . 12 ( = (𝑓𝑔) → (𝐺) = ((𝑓𝑔) ∘ 𝐺))
6766eqeq2d 2781 . . . . . . . . . . 11 ( = (𝑓𝑔) → (𝐺 = (𝐺) ↔ 𝐺 = ((𝑓𝑔) ∘ 𝐺)))
68 coeq1 5418 . . . . . . . . . . . 12 ( = ( I ↾ 𝑌) → (𝐺) = (( I ↾ 𝑌) ∘ 𝐺))
6968eqeq2d 2781 . . . . . . . . . . 11 ( = ( I ↾ 𝑌) → (𝐺 = (𝐺) ↔ 𝐺 = (( I ↾ 𝑌) ∘ 𝐺)))
70 simpr3 1237 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐺𝑥) = (𝐺𝑦))) → (𝐺𝑥) = (𝐺𝑦))
711, 3, 7, 70qtopeu 21740 . . . . . . . . . . . 12 (𝜑 → ∃! ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺))𝐺 = (𝐺))
7271ad2antrr 705 . . . . . . . . . . 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 21291 . . . . . . . . . . . 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 573 . . . . . . . . . . 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 21282 . . . . . . . . . . . . 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 705 . . . . . . . . . . 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 5423 . . . . . . . . . . . . 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 2805 . . . . . . . . . . . 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 5798 . . . . . . . . . . . 12 ((𝑓𝑔) ∘ 𝐺) = (𝑓 ∘ (𝑔𝐺))
8179, 80syl6eqr 2823 . . . . . . . . . . 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 6256 . . . . . . . . . . . . . . 15 (𝐺:𝑋onto𝑌𝐺:𝑋𝑌)
833, 82syl 17 . . . . . . . . . . . . . 14 (𝜑𝐺:𝑋𝑌)
8483ad2antrr 705 . . . . . . . . . . . . 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 6219 . . . . . . . . . . . . 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 2777 . . . . . . . . . . 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 3555 . . . . . . . . . 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 6693 . . . . . . . . 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 2850 . . . . . . . 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 3183 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
92 ishmeo 21783 . . . . . . 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 572 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))
94 simprr 756 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝐺 = (𝑓𝐹))
9593, 94jca 501 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹)))
9695ex 397 . . . 4 (𝜑 → ((𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹)) → (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))))
9796reximdv2 3162 . . 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 2791 . . . 4 ((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → (𝑓𝐹) = (𝑔𝐹))
1002adantr 466 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝐹:𝑋onto𝑌)
10130adantr 466 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
10233adantr 466 . . . . . . 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 754 . . . . . . 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 21787 . . . . . . 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 1476 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑓:𝑌1-1-onto𝑌)
106 f1ofn 6279 . . . . . 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 756 . . . . . . 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 21787 . . . . . . 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 1476 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑔:𝑌1-1-onto𝑌)
111 f1ofn 6279 . . . . . 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 6690 . . . . 5 ((𝐹:𝑋onto𝑌𝑓 Fn 𝑌𝑔 Fn 𝑌) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
114100, 107, 112, 113syl3anc 1476 . . . 4 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
11599, 114syl5ib 234 . . 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 3120 . 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 5418 . . . 4 (𝑓 = 𝑔 → (𝑓𝐹) = (𝑔𝐹))
118117eqeq2d 2781 . . 3 (𝑓 = 𝑔 → (𝐺 = (𝑓𝐹) ↔ 𝐺 = (𝑔𝐹)))
119118reu4 3552 . 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 572 1 (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  ∃!wreu 3063   I cid 5156  ccnv 5248  cres 5251  ccom 5253   Fn wfn 6026  wf 6027  ontowfo 6029  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793   qTop cqtop 16371  TopOnctopon 20935   Cn ccn 21249  Homeochmeo 21777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-qtop 16375  df-top 20919  df-topon 20936  df-cn 21252  df-hmeo 21779
This theorem is referenced by: (None)
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