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Theorem qtopeu 21848
Description: Universal property of the quotient topology. If 𝐺 is a function from 𝐽 to 𝐾 which is equal on all equivalent elements under 𝐹, then there is a unique continuous map 𝑓:(𝐽 / 𝐹)⟶𝐾 such that 𝐺 = 𝑓𝐹, and we say that 𝐺 "passes to the quotient". (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypotheses
Ref Expression
qtopeu.1 (𝜑𝐽 ∈ (TopOn‘𝑋))
qtopeu.3 (𝜑𝐹:𝑋onto𝑌)
qtopeu.4 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
qtopeu.5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐺𝑥) = (𝐺𝑦))
Assertion
Ref Expression
qtopeu (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐹   𝑓,𝐽,𝑥   𝑓,𝐾,𝑥   𝑥,𝑋,𝑦   𝑓,𝐺,𝑥,𝑦   𝜑,𝑓,𝑥,𝑦   𝑓,𝑌,𝑥
Allowed substitution hints:   𝐽(𝑦)   𝐾(𝑦)   𝑋(𝑓)   𝑌(𝑦)

Proof of Theorem qtopeu
Dummy variables 𝑔 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qtopeu.3 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑋onto𝑌)
2 fofn 6333 . . . . . . . . . . . . . . . 16 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
31, 2syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝑋)
43adantr 473 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → 𝐹 Fn 𝑋)
5 fniniseg 6564 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑋 → (𝑦 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
64, 5syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (𝑦 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
7 eqcom 2806 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
873anbi3i 1199 . . . . . . . . . . . . . . . . 17 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)))
9 3anass 1117 . . . . . . . . . . . . . . . . 17 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)) ↔ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
108, 9bitri 267 . . . . . . . . . . . . . . . 16 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
11 qtopeu.5 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐺𝑥) = (𝐺𝑦))
1210, 11sylan2br 589 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)))) → (𝐺𝑥) = (𝐺𝑦))
1312eqcomd 2805 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)))) → (𝐺𝑦) = (𝐺𝑥))
1413expr 449 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → ((𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)) → (𝐺𝑦) = (𝐺𝑥)))
156, 14sylbid 232 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (𝑦 ∈ (𝐹 “ {(𝐹𝑥)}) → (𝐺𝑦) = (𝐺𝑥)))
1615ralrimiv 3146 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → ∀𝑦 ∈ (𝐹 “ {(𝐹𝑥)})(𝐺𝑦) = (𝐺𝑥))
17 qtopeu.1 . . . . . . . . . . . . . . 15 (𝜑𝐽 ∈ (TopOn‘𝑋))
18 qtopeu.4 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
19 cntop2 21374 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2018, 19syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ Top)
21 eqid 2799 . . . . . . . . . . . . . . . . 17 𝐾 = 𝐾
2221toptopon 21050 . . . . . . . . . . . . . . . 16 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
2320, 22sylib 210 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
24 cnf2 21382 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐺 ∈ (𝐽 Cn 𝐾)) → 𝐺:𝑋 𝐾)
2517, 23, 18, 24syl3anc 1491 . . . . . . . . . . . . . 14 (𝜑𝐺:𝑋 𝐾)
2625ffnd 6257 . . . . . . . . . . . . 13 (𝜑𝐺 Fn 𝑋)
2726adantr 473 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → 𝐺 Fn 𝑋)
28 cnvimass 5702 . . . . . . . . . . . . 13 (𝐹 “ {(𝐹𝑥)}) ⊆ dom 𝐹
29 fof 6331 . . . . . . . . . . . . . . . 16 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
301, 29syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑋𝑌)
3130fdmd 6265 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐹 = 𝑋)
3231adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → dom 𝐹 = 𝑋)
3328, 32syl5sseq 3849 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (𝐹 “ {(𝐹𝑥)}) ⊆ 𝑋)
34 eqeq1 2803 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑦) → (𝑤 = (𝐺𝑥) ↔ (𝐺𝑦) = (𝐺𝑥)))
3534ralima 6727 . . . . . . . . . . . 12 ((𝐺 Fn 𝑋 ∧ (𝐹 “ {(𝐹𝑥)}) ⊆ 𝑋) → (∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥) ↔ ∀𝑦 ∈ (𝐹 “ {(𝐹𝑥)})(𝐺𝑦) = (𝐺𝑥)))
3627, 33, 35syl2anc 580 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥) ↔ ∀𝑦 ∈ (𝐹 “ {(𝐹𝑥)})(𝐺𝑦) = (𝐺𝑥)))
3716, 36mpbird 249 . . . . . . . . . 10 ((𝜑𝑥𝑋) → ∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥))
3825fdmd 6265 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝐺 = 𝑋)
3938eleq2d 2864 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ dom 𝐺𝑥𝑋))
4039biimpar 470 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → 𝑥 ∈ dom 𝐺)
41 simpr 478 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → 𝑥𝑋)
42 eqidd 2800 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → (𝐹𝑥) = (𝐹𝑥))
43 fniniseg 6564 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝑋 → (𝑥 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑥𝑋 ∧ (𝐹𝑥) = (𝐹𝑥))))
444, 43syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → (𝑥 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑥𝑋 ∧ (𝐹𝑥) = (𝐹𝑥))))
4541, 42, 44mpbir2and 705 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → 𝑥 ∈ (𝐹 “ {(𝐹𝑥)}))
46 inelcm 4227 . . . . . . . . . . . . 13 ((𝑥 ∈ dom 𝐺𝑥 ∈ (𝐹 “ {(𝐹𝑥)})) → (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
4740, 45, 46syl2anc 580 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
48 imadisj 5701 . . . . . . . . . . . . 13 ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) = ∅ ↔ (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) = ∅)
4948necon3bii 3023 . . . . . . . . . . . 12 ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) ≠ ∅ ↔ (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
5047, 49sylibr 226 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
51 eqsn 4548 . . . . . . . . . . 11 ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) ≠ ∅ → ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)} ↔ ∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥)))
5250, 51syl 17 . . . . . . . . . 10 ((𝜑𝑥𝑋) → ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)} ↔ ∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥)))
5337, 52mpbird 249 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)})
5453unieqd 4638 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)})
55 fvex 6424 . . . . . . . . 9 (𝐺𝑥) ∈ V
5655unisn 4644 . . . . . . . 8 {(𝐺𝑥)} = (𝐺𝑥)
5754, 56syl6req 2850 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐺𝑥) = (𝐺 “ (𝐹 “ {(𝐹𝑥)})))
5857mpteq2dva 4937 . . . . . 6 (𝜑 → (𝑥𝑋 ↦ (𝐺𝑥)) = (𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)}))))
5925feqmptd 6474 . . . . . 6 (𝜑𝐺 = (𝑥𝑋 ↦ (𝐺𝑥)))
6030ffvelrnda 6585 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ 𝑌)
6130feqmptd 6474 . . . . . . 7 (𝜑𝐹 = (𝑥𝑋 ↦ (𝐹𝑥)))
62 eqidd 2800 . . . . . . 7 (𝜑 → (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) = (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))))
63 sneq 4378 . . . . . . . . . 10 (𝑤 = (𝐹𝑥) → {𝑤} = {(𝐹𝑥)})
6463imaeq2d 5683 . . . . . . . . 9 (𝑤 = (𝐹𝑥) → (𝐹 “ {𝑤}) = (𝐹 “ {(𝐹𝑥)}))
6564imaeq2d 5683 . . . . . . . 8 (𝑤 = (𝐹𝑥) → (𝐺 “ (𝐹 “ {𝑤})) = (𝐺 “ (𝐹 “ {(𝐹𝑥)})))
6665unieqd 4638 . . . . . . 7 (𝑤 = (𝐹𝑥) → (𝐺 “ (𝐹 “ {𝑤})) = (𝐺 “ (𝐹 “ {(𝐹𝑥)})))
6760, 61, 62, 66fmptco 6623 . . . . . 6 (𝜑 → ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹) = (𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)}))))
6858, 59, 673eqtr4d 2843 . . . . 5 (𝜑𝐺 = ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹))
6968, 18eqeltrrd 2879 . . . 4 (𝜑 → ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹) ∈ (𝐽 Cn 𝐾))
7025ffvelrnda 6585 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝐺𝑥) ∈ 𝐾)
7157, 70eqeltrrd 2879 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾)
7271ralrimiva 3147 . . . . . . 7 (𝜑 → ∀𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾)
7366eqcomd 2805 . . . . . . . . . . 11 (𝑤 = (𝐹𝑥) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = (𝐺 “ (𝐹 “ {𝑤})))
7473eqcoms 2807 . . . . . . . . . 10 ((𝐹𝑥) = 𝑤 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = (𝐺 “ (𝐹 “ {𝑤})))
7574eleq1d 2863 . . . . . . . . 9 ((𝐹𝑥) = 𝑤 → ( (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾))
7675cbvfo 6772 . . . . . . . 8 (𝐹:𝑋onto𝑌 → (∀𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾 ↔ ∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾))
771, 76syl 17 . . . . . . 7 (𝜑 → (∀𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾 ↔ ∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾))
7872, 77mpbid 224 . . . . . 6 (𝜑 → ∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾)
79 eqid 2799 . . . . . . 7 (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) = (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})))
8079fmpt 6606 . . . . . 6 (∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾 ↔ (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))):𝑌 𝐾)
8178, 80sylib 210 . . . . 5 (𝜑 → (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))):𝑌 𝐾)
82 qtopcn 21846 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) ∧ (𝐹:𝑋onto𝑌 ∧ (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))):𝑌 𝐾)) → ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹) ∈ (𝐽 Cn 𝐾)))
8317, 23, 1, 81, 82syl22anc 868 . . . 4 (𝜑 → ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹) ∈ (𝐽 Cn 𝐾)))
8469, 83mpbird 249 . . 3 (𝜑 → (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
85 coeq1 5483 . . . 4 (𝑓 = (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) → (𝑓𝐹) = ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹))
8685rspceeqv 3515 . . 3 (((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝐺 = ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹)) → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹))
8784, 68, 86syl2anc 580 . 2 (𝜑 → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹))
88 eqtr2 2819 . . . 4 ((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → (𝑓𝐹) = (𝑔𝐹))
891adantr 473 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝐹:𝑋onto𝑌)
90 qtoptopon 21836 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
9117, 1, 90syl2anc 580 . . . . . . . 8 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
9291adantr 473 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
9323adantr 473 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝐾 ∈ (TopOn‘ 𝐾))
94 simprl 788 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
95 cnf2 21382 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)) → 𝑓:𝑌 𝐾)
9692, 93, 94, 95syl3anc 1491 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓:𝑌 𝐾)
9796ffnd 6257 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓 Fn 𝑌)
98 simprr 790 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
99 cnf2 21382 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)) → 𝑔:𝑌 𝐾)
10092, 93, 98, 99syl3anc 1491 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔:𝑌 𝐾)
101100ffnd 6257 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔 Fn 𝑌)
102 cocan2 6775 . . . . 5 ((𝐹:𝑋onto𝑌𝑓 Fn 𝑌𝑔 Fn 𝑌) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
10389, 97, 101, 102syl3anc 1491 . . . 4 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
10488, 103syl5ib 236 . . 3 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → ((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔))
105104ralrimivva 3152 . 2 (𝜑 → ∀𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)∀𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔))
106 coeq1 5483 . . . 4 (𝑓 = 𝑔 → (𝑓𝐹) = (𝑔𝐹))
107106eqeq2d 2809 . . 3 (𝑓 = 𝑔 → (𝐺 = (𝑓𝐹) ↔ 𝐺 = (𝑔𝐹)))
108107reu4 3596 . 2 (∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹) ↔ (∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹) ∧ ∀𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)∀𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔)))
10987, 105, 108sylanbrc 579 1 (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wral 3089  wrex 3090  ∃!wreu 3091  cin 3768  wss 3769  c0 4115  {csn 4368   cuni 4628  cmpt 4922  ccnv 5311  dom cdm 5312  cima 5315  ccom 5316   Fn wfn 6096  wf 6097  ontowfo 6099  cfv 6101  (class class class)co 6878   qTop cqtop 16478  Topctop 21026  TopOnctopon 21043   Cn ccn 21357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-qtop 16482  df-top 21027  df-topon 21044  df-cn 21360
This theorem is referenced by:  qtophmeo  21949
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