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Theorem qtophmeo 23168
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 6758 . . . . . . 7 (𝐺:𝑋onto𝑌𝐺 Fn 𝑋)
53, 4syl 17 . . . . . 6 (𝜑𝐺 Fn 𝑋)
6 qtopid 23056 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺 Fn 𝑋) → 𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
71, 5, 6syl2anc 584 . . . . 5 (𝜑𝐺 ∈ (𝐽 Cn (𝐽 qTop 𝐺)))
8 df-3an 1089 . . . . . 6 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦)))
9 qtophmeo.5 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐺𝑥) = (𝐺𝑦)))
109biimpd 228 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) → (𝐺𝑥) = (𝐺𝑦)))
1110impr 455 . . . . . 6 ((𝜑 ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐺𝑥) = (𝐺𝑦))
128, 11sylan2b 594 . . . . 5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐺𝑥) = (𝐺𝑦))
131, 2, 7, 12qtopeu 23067 . . . 4 (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))
14 reurex 3357 . . . 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 769 . . . 4 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)))
17 fofn 6758 . . . . . . . . . 10 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
182, 17syl 17 . . . . . . . . 9 (𝜑𝐹 Fn 𝑋)
19 qtopid 23056 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
201, 18, 19syl2anc 584 . . . . . . . 8 (𝜑𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
21 df-3an 1089 . . . . . . . . 9 ((𝑥𝑋𝑦𝑋 ∧ (𝐺𝑥) = (𝐺𝑦)) ↔ ((𝑥𝑋𝑦𝑋) ∧ (𝐺𝑥) = (𝐺𝑦)))
229biimprd 247 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐺𝑥) = (𝐺𝑦) → (𝐹𝑥) = (𝐹𝑦)))
2322impr 455 . . . . . . . . 9 ((𝜑 ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐺𝑥) = (𝐺𝑦))) → (𝐹𝑥) = (𝐹𝑦))
2421, 23sylan2b 594 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐺𝑥) = (𝐺𝑦))) → (𝐹𝑥) = (𝐹𝑦))
251, 3, 20, 24qtopeu 23067 . . . . . . 7 (𝜑 → ∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺))
2625adantr 481 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → ∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺))
27 reurex 3357 . . . . . 6 (∃!𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺) → ∃𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺))
2826, 27syl 17 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → ∃𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))𝐹 = (𝑔𝐺))
29 qtoptopon 23055 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
301, 2, 29syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
3130ad2antrr 724 . . . . . . . 8 (((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 23055 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐺:𝑋onto𝑌) → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
331, 3, 32syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌))
3433ad2antrr 724 . . . . . . . 8 (((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 775 . . . . . . . 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 𝐹) Cn (𝐽 qTop 𝐺)))
36 cnf2 22600 . . . . . . . 8 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺))) → 𝑓:𝑌𝑌)
3731, 34, 35, 36syl3anc 1371 . . . . . . 7 (((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 769 . . . . . . . 8 (((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 22600 . . . . . . . 8 (((𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹))) → 𝑔:𝑌𝑌)
4034, 31, 38, 39syl3anc 1371 . . . . . . 7 (((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 5813 . . . . . . . . 9 ( = (𝑔𝑓) → (𝐹) = ((𝑔𝑓) ∘ 𝐹))
4241eqeq2d 2747 . . . . . . . 8 ( = (𝑔𝑓) → (𝐹 = (𝐹) ↔ 𝐹 = ((𝑔𝑓) ∘ 𝐹)))
43 coeq1 5813 . . . . . . . . 9 ( = ( I ↾ 𝑌) → (𝐹) = (( I ↾ 𝑌) ∘ 𝐹))
4443eqeq2d 2747 . . . . . . . 8 ( = ( I ↾ 𝑌) → (𝐹 = (𝐹) ↔ 𝐹 = (( I ↾ 𝑌) ∘ 𝐹)))
45 simpr3 1196 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
461, 2, 20, 45qtopeu 23067 . . . . . . . . 9 (𝜑 → ∃! ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹))𝐹 = (𝐹))
4746ad2antrr 724 . . . . . . . 8 (((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 22617 . . . . . . . . 9 ((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 584 . . . . . . . 8 (((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 22608 . . . . . . . . . 10 ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
5130, 50syl 17 . . . . . . . . 9 (𝜑 → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐹)))
5251ad2antrr 724 . . . . . . . 8 (((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 771 . . . . . . . . . 10 (((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 776 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) ∧ (𝑔 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)) ∧ 𝐹 = (𝑔𝐺))) → 𝐺 = (𝑓𝐹))
5554coeq2d 5818 . . . . . . . . . 10 (((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 2776 . . . . . . . . 9 (((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 6217 . . . . . . . . 9 ((𝑔𝑓) ∘ 𝐹) = (𝑔 ∘ (𝑓𝐹))
5856, 57eqtr4di 2794 . . . . . . . 8 (((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 6756 . . . . . . . . . . . 12 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
602, 59syl 17 . . . . . . . . . . 11 (𝜑𝐹:𝑋𝑌)
6160ad2antrr 724 . . . . . . . . . 10 (((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 6717 . . . . . . . . . 10 (𝐹:𝑋𝑌 → (( I ↾ 𝑌) ∘ 𝐹) = 𝐹)
6361, 62syl 17 . . . . . . . . 9 (((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 2742 . . . . . . . 8 (((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 3694 . . . . . . 7 (((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 5813 . . . . . . . . 9 ( = (𝑓𝑔) → (𝐺) = ((𝑓𝑔) ∘ 𝐺))
6766eqeq2d 2747 . . . . . . . 8 ( = (𝑓𝑔) → (𝐺 = (𝐺) ↔ 𝐺 = ((𝑓𝑔) ∘ 𝐺)))
68 coeq1 5813 . . . . . . . . 9 ( = ( I ↾ 𝑌) → (𝐺) = (( I ↾ 𝑌) ∘ 𝐺))
6968eqeq2d 2747 . . . . . . . 8 ( = ( I ↾ 𝑌) → (𝐺 = (𝐺) ↔ 𝐺 = (( I ↾ 𝑌) ∘ 𝐺)))
70 simpr3 1196 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐺𝑥) = (𝐺𝑦))) → (𝐺𝑥) = (𝐺𝑦))
711, 3, 7, 70qtopeu 23067 . . . . . . . . 9 (𝜑 → ∃! ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺))𝐺 = (𝐺))
7271ad2antrr 724 . . . . . . . 8 (((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 22617 . . . . . . . . 9 ((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 584 . . . . . . . 8 (((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 22608 . . . . . . . . . 10 ((𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
7633, 75syl 17 . . . . . . . . 9 (𝜑 → ( I ↾ 𝑌) ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐺)))
7776ad2antrr 724 . . . . . . . 8 (((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 5818 . . . . . . . . . 10 (((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 2776 . . . . . . . . 9 (((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 6217 . . . . . . . . 9 ((𝑓𝑔) ∘ 𝐺) = (𝑓 ∘ (𝑔𝐺))
8179, 80eqtr4di 2794 . . . . . . . 8 (((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 6756 . . . . . . . . . . . 12 (𝐺:𝑋onto𝑌𝐺:𝑋𝑌)
833, 82syl 17 . . . . . . . . . . 11 (𝜑𝐺:𝑋𝑌)
8483ad2antrr 724 . . . . . . . . . 10 (((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 6717 . . . . . . . . . 10 (𝐺:𝑋𝑌 → (( I ↾ 𝑌) ∘ 𝐺) = 𝐺)
8684, 85syl 17 . . . . . . . . 9 (((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 2742 . . . . . . . 8 (((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 3694 . . . . . . 7 (((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 7241 . . . . . 6 (((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 2838 . . . . 5 (((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 3158 . . . 4 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐺) Cn (𝐽 qTop 𝐹)))
92 ishmeo 23110 . . . 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 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))
94 simprr 771 . . 3 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn (𝐽 qTop 𝐺)) ∧ 𝐺 = (𝑓𝐹))) → 𝐺 = (𝑓𝐹))
9515, 93, 94reximssdv 3169 . 2 (𝜑 → ∃𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))
96 eqtr2 2760 . . . 4 ((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → (𝑓𝐹) = (𝑔𝐹))
972adantr 481 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝐹:𝑋onto𝑌)
9830adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
9933adantr 481 . . . . . . 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 769 . . . . . . 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 23114 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))) → 𝑓:𝑌1-1-onto𝑌)
10298, 99, 100, 101syl3anc 1371 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑓:𝑌1-1-onto𝑌)
103 f1ofn 6785 . . . . . 6 (𝑓:𝑌1-1-onto𝑌𝑓 Fn 𝑌)
104102, 103syl 17 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑓 Fn 𝑌)
105 simprr 771 . . . . . . 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 23114 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ (𝐽 qTop 𝐺) ∈ (TopOn‘𝑌) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))) → 𝑔:𝑌1-1-onto𝑌)
10798, 99, 105, 106syl3anc 1371 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑔:𝑌1-1-onto𝑌)
108 f1ofn 6785 . . . . . 6 (𝑔:𝑌1-1-onto𝑌𝑔 Fn 𝑌)
109107, 108syl 17 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → 𝑔 Fn 𝑌)
110 cocan2 7238 . . . . 5 ((𝐹:𝑋onto𝑌𝑓 Fn 𝑌𝑔 Fn 𝑌) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
11197, 104, 109, 110syl3anc 1371 . . . 4 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
11296, 111imbitrid 243 . . 3 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺)))) → ((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔))
113112ralrimivva 3197 . 2 (𝜑 → ∀𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))∀𝑔 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔))
114 coeq1 5813 . . . 4 (𝑓 = 𝑔 → (𝑓𝐹) = (𝑔𝐹))
115114eqeq2d 2747 . . 3 (𝑓 = 𝑔 → (𝐺 = (𝑓𝐹) ↔ 𝐺 = (𝑔𝐹)))
116115reu4 3689 . 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)))
11795, 113, 116sylanbrc 583 1 (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  ∃!wreu 3351   I cid 5530  ccnv 5632  cres 5635  ccom 5637   Fn wfn 6491  wf 6492  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357   qTop cqtop 17385  TopOnctopon 22259   Cn ccn 22575  Homeochmeo 23104
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-qtop 17389  df-top 22243  df-topon 22260  df-cn 22578  df-hmeo 23106
This theorem is referenced by: (None)
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