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Theorem qtopeu 23664
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 6812 . . . . . . . . . . . . . . . 16 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
31, 2syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝑋)
43adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → 𝐹 Fn 𝑋)
5 fniniseg 7068 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑋 → (𝑦 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
64, 5syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (𝑦 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
7 eqcom 2732 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
873anbi3i 1156 . . . . . . . . . . . . . . . . 17 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)))
9 3anass 1092 . . . . . . . . . . . . . . . . 17 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)) ↔ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
108, 9bitri 274 . . . . . . . . . . . . . . . 16 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
11 qtopeu.5 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐺𝑥) = (𝐺𝑦))
1210, 11sylan2br 593 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)))) → (𝐺𝑥) = (𝐺𝑦))
1312eqcomd 2731 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)))) → (𝐺𝑦) = (𝐺𝑥))
1413expr 455 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → ((𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)) → (𝐺𝑦) = (𝐺𝑥)))
156, 14sylbid 239 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (𝑦 ∈ (𝐹 “ {(𝐹𝑥)}) → (𝐺𝑦) = (𝐺𝑥)))
1615ralrimiv 3134 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → ∀𝑦 ∈ (𝐹 “ {(𝐹𝑥)})(𝐺𝑦) = (𝐺𝑥))
17 qtopeu.1 . . . . . . . . . . . . . . 15 (𝜑𝐽 ∈ (TopOn‘𝑋))
18 qtopeu.4 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
19 cntop2 23189 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2018, 19syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ Top)
21 toptopon2 22864 . . . . . . . . . . . . . . . 16 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
2220, 21sylib 217 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
23 cnf2 23197 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐺 ∈ (𝐽 Cn 𝐾)) → 𝐺:𝑋 𝐾)
2417, 22, 18, 23syl3anc 1368 . . . . . . . . . . . . . 14 (𝜑𝐺:𝑋 𝐾)
2524ffnd 6724 . . . . . . . . . . . . 13 (𝜑𝐺 Fn 𝑋)
2625adantr 479 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → 𝐺 Fn 𝑋)
27 cnvimass 6086 . . . . . . . . . . . . 13 (𝐹 “ {(𝐹𝑥)}) ⊆ dom 𝐹
28 fof 6810 . . . . . . . . . . . . . . . 16 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
291, 28syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑋𝑌)
3029fdmd 6733 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐹 = 𝑋)
3130adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → dom 𝐹 = 𝑋)
3227, 31sseqtrid 4029 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (𝐹 “ {(𝐹𝑥)}) ⊆ 𝑋)
33 eqeq1 2729 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑦) → (𝑤 = (𝐺𝑥) ↔ (𝐺𝑦) = (𝐺𝑥)))
3433ralima 7250 . . . . . . . . . . . 12 ((𝐺 Fn 𝑋 ∧ (𝐹 “ {(𝐹𝑥)}) ⊆ 𝑋) → (∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥) ↔ ∀𝑦 ∈ (𝐹 “ {(𝐹𝑥)})(𝐺𝑦) = (𝐺𝑥)))
3526, 32, 34syl2anc 582 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥) ↔ ∀𝑦 ∈ (𝐹 “ {(𝐹𝑥)})(𝐺𝑦) = (𝐺𝑥)))
3616, 35mpbird 256 . . . . . . . . . 10 ((𝜑𝑥𝑋) → ∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥))
3724fdmd 6733 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝐺 = 𝑋)
3837eleq2d 2811 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ dom 𝐺𝑥𝑋))
3938biimpar 476 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → 𝑥 ∈ dom 𝐺)
40 simpr 483 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → 𝑥𝑋)
41 eqidd 2726 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → (𝐹𝑥) = (𝐹𝑥))
42 fniniseg 7068 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝑋 → (𝑥 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑥𝑋 ∧ (𝐹𝑥) = (𝐹𝑥))))
434, 42syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → (𝑥 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑥𝑋 ∧ (𝐹𝑥) = (𝐹𝑥))))
4440, 41, 43mpbir2and 711 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → 𝑥 ∈ (𝐹 “ {(𝐹𝑥)}))
45 inelcm 4466 . . . . . . . . . . . . 13 ((𝑥 ∈ dom 𝐺𝑥 ∈ (𝐹 “ {(𝐹𝑥)})) → (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
4639, 44, 45syl2anc 582 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
47 imadisj 6084 . . . . . . . . . . . . 13 ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) = ∅ ↔ (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) = ∅)
4847necon3bii 2982 . . . . . . . . . . . 12 ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) ≠ ∅ ↔ (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
4946, 48sylibr 233 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
50 eqsn 4834 . . . . . . . . . . 11 ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) ≠ ∅ → ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)} ↔ ∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥)))
5149, 50syl 17 . . . . . . . . . 10 ((𝜑𝑥𝑋) → ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)} ↔ ∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥)))
5236, 51mpbird 256 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)})
5352unieqd 4922 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)})
54 fvex 6909 . . . . . . . . 9 (𝐺𝑥) ∈ V
5554unisn 4930 . . . . . . . 8 {(𝐺𝑥)} = (𝐺𝑥)
5653, 55eqtr2di 2782 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐺𝑥) = (𝐺 “ (𝐹 “ {(𝐹𝑥)})))
5756mpteq2dva 5249 . . . . . 6 (𝜑 → (𝑥𝑋 ↦ (𝐺𝑥)) = (𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)}))))
5824feqmptd 6966 . . . . . 6 (𝜑𝐺 = (𝑥𝑋 ↦ (𝐺𝑥)))
5929ffvelcdmda 7093 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ 𝑌)
6029feqmptd 6966 . . . . . . 7 (𝜑𝐹 = (𝑥𝑋 ↦ (𝐹𝑥)))
61 eqidd 2726 . . . . . . 7 (𝜑 → (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) = (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))))
62 sneq 4640 . . . . . . . . . 10 (𝑤 = (𝐹𝑥) → {𝑤} = {(𝐹𝑥)})
6362imaeq2d 6064 . . . . . . . . 9 (𝑤 = (𝐹𝑥) → (𝐹 “ {𝑤}) = (𝐹 “ {(𝐹𝑥)}))
6463imaeq2d 6064 . . . . . . . 8 (𝑤 = (𝐹𝑥) → (𝐺 “ (𝐹 “ {𝑤})) = (𝐺 “ (𝐹 “ {(𝐹𝑥)})))
6564unieqd 4922 . . . . . . 7 (𝑤 = (𝐹𝑥) → (𝐺 “ (𝐹 “ {𝑤})) = (𝐺 “ (𝐹 “ {(𝐹𝑥)})))
6659, 60, 61, 65fmptco 7138 . . . . . 6 (𝜑 → ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹) = (𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)}))))
6757, 58, 663eqtr4d 2775 . . . . 5 (𝜑𝐺 = ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹))
6867, 18eqeltrrd 2826 . . . 4 (𝜑 → ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹) ∈ (𝐽 Cn 𝐾))
6924ffvelcdmda 7093 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝐺𝑥) ∈ 𝐾)
7056, 69eqeltrrd 2826 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾)
7170ralrimiva 3135 . . . . . . 7 (𝜑 → ∀𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾)
7265eqcomd 2731 . . . . . . . . . . 11 (𝑤 = (𝐹𝑥) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = (𝐺 “ (𝐹 “ {𝑤})))
7372eqcoms 2733 . . . . . . . . . 10 ((𝐹𝑥) = 𝑤 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = (𝐺 “ (𝐹 “ {𝑤})))
7473eleq1d 2810 . . . . . . . . 9 ((𝐹𝑥) = 𝑤 → ( (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾))
7574cbvfo 7298 . . . . . . . 8 (𝐹:𝑋onto𝑌 → (∀𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾 ↔ ∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾))
761, 75syl 17 . . . . . . 7 (𝜑 → (∀𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾 ↔ ∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾))
7771, 76mpbid 231 . . . . . 6 (𝜑 → ∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾)
78 eqid 2725 . . . . . . 7 (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) = (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})))
7978fmpt 7119 . . . . . 6 (∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾 ↔ (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))):𝑌 𝐾)
8077, 79sylib 217 . . . . 5 (𝜑 → (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))):𝑌 𝐾)
81 qtopcn 23662 . . . . 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)))
8217, 22, 1, 80, 81syl22anc 837 . . . 4 (𝜑 → ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹) ∈ (𝐽 Cn 𝐾)))
8368, 82mpbird 256 . . 3 (𝜑 → (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
84 coeq1 5860 . . . 4 (𝑓 = (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) → (𝑓𝐹) = ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹))
8584rspceeqv 3628 . . 3 (((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝐺 = ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹)) → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹))
8683, 67, 85syl2anc 582 . 2 (𝜑 → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹))
87 eqtr2 2749 . . . 4 ((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → (𝑓𝐹) = (𝑔𝐹))
881adantr 479 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝐹:𝑋onto𝑌)
89 qtoptopon 23652 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
9017, 1, 89syl2anc 582 . . . . . . . 8 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
9190adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
9222adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝐾 ∈ (TopOn‘ 𝐾))
93 simprl 769 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
94 cnf2 23197 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)) → 𝑓:𝑌 𝐾)
9591, 92, 93, 94syl3anc 1368 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓:𝑌 𝐾)
9695ffnd 6724 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓 Fn 𝑌)
97 simprr 771 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
98 cnf2 23197 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)) → 𝑔:𝑌 𝐾)
9991, 92, 97, 98syl3anc 1368 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔:𝑌 𝐾)
10099ffnd 6724 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔 Fn 𝑌)
101 cocan2 7301 . . . . 5 ((𝐹:𝑋onto𝑌𝑓 Fn 𝑌𝑔 Fn 𝑌) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
10288, 96, 100, 101syl3anc 1368 . . . 4 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
10387, 102imbitrid 243 . . 3 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → ((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔))
104103ralrimivva 3190 . 2 (𝜑 → ∀𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)∀𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔))
105 coeq1 5860 . . . 4 (𝑓 = 𝑔 → (𝑓𝐹) = (𝑔𝐹))
106105eqeq2d 2736 . . 3 (𝑓 = 𝑔 → (𝐺 = (𝑓𝐹) ↔ 𝐺 = (𝑔𝐹)))
107106reu4 3723 . 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)))
10886, 104, 107sylanbrc 581 1 (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  ∃!wreu 3361  cin 3943  wss 3944  c0 4322  {csn 4630   cuni 4909  cmpt 5232  ccnv 5677  dom cdm 5678  cima 5681  ccom 5682   Fn wfn 6544  wf 6545  ontowfo 6547  cfv 6549  (class class class)co 7419   qTop cqtop 17488  Topctop 22839  TopOnctopon 22856   Cn ccn 23172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-qtop 17492  df-top 22840  df-topon 22857  df-cn 23175
This theorem is referenced by:  qtophmeo  23765
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