Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  qtopeu Structured version   Visualization version   GIF version

Theorem qtopeu 21329
 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 = 𝑓 ∘ 𝐹, 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 6030 . . . . . . . . . . . . . . . 16 (𝐹:𝑋onto𝑌𝐹 Fn 𝑋)
31, 2syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝑋)
43adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → 𝐹 Fn 𝑋)
5 fniniseg 6246 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑋 → (𝑦 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
64, 5syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (𝑦 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
7 eqcom 2617 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
873anbi3i 1248 . . . . . . . . . . . . . . . . 17 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)))
9 3anass 1035 . . . . . . . . . . . . . . . . 17 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)) ↔ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
108, 9bitri 263 . . . . . . . . . . . . . . . 16 ((𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦)) ↔ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥))))
11 qtopeu.5 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑋𝑦𝑋 ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐺𝑥) = (𝐺𝑦))
1210, 11sylan2br 492 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)))) → (𝐺𝑥) = (𝐺𝑦))
1312eqcomd 2616 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑋 ∧ (𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)))) → (𝐺𝑦) = (𝐺𝑥))
1413expr 641 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → ((𝑦𝑋 ∧ (𝐹𝑦) = (𝐹𝑥)) → (𝐺𝑦) = (𝐺𝑥)))
156, 14sylbid 229 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (𝑦 ∈ (𝐹 “ {(𝐹𝑥)}) → (𝐺𝑦) = (𝐺𝑥)))
1615ralrimiv 2948 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → ∀𝑦 ∈ (𝐹 “ {(𝐹𝑥)})(𝐺𝑦) = (𝐺𝑥))
17 qtopeu.1 . . . . . . . . . . . . . . 15 (𝜑𝐽 ∈ (TopOn‘𝑋))
18 qtopeu.4 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
19 cntop2 20855 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2018, 19syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ Top)
21 eqid 2610 . . . . . . . . . . . . . . . . 17 𝐾 = 𝐾
2221toptopon 20548 . . . . . . . . . . . . . . . 16 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
2320, 22sylib 207 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
24 cnf2 20863 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐺 ∈ (𝐽 Cn 𝐾)) → 𝐺:𝑋 𝐾)
2517, 23, 18, 24syl3anc 1318 . . . . . . . . . . . . . 14 (𝜑𝐺:𝑋 𝐾)
26 ffn 5958 . . . . . . . . . . . . . 14 (𝐺:𝑋 𝐾𝐺 Fn 𝑋)
2725, 26syl 17 . . . . . . . . . . . . 13 (𝜑𝐺 Fn 𝑋)
2827adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → 𝐺 Fn 𝑋)
29 cnvimass 5404 . . . . . . . . . . . . 13 (𝐹 “ {(𝐹𝑥)}) ⊆ dom 𝐹
30 fof 6028 . . . . . . . . . . . . . . . 16 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
311, 30syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑋𝑌)
32 fdm 5964 . . . . . . . . . . . . . . 15 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
3331, 32syl 17 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐹 = 𝑋)
3433adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → dom 𝐹 = 𝑋)
3529, 34syl5sseq 3616 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (𝐹 “ {(𝐹𝑥)}) ⊆ 𝑋)
36 eqeq1 2614 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑦) → (𝑤 = (𝐺𝑥) ↔ (𝐺𝑦) = (𝐺𝑥)))
3736ralima 6402 . . . . . . . . . . . 12 ((𝐺 Fn 𝑋 ∧ (𝐹 “ {(𝐹𝑥)}) ⊆ 𝑋) → (∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥) ↔ ∀𝑦 ∈ (𝐹 “ {(𝐹𝑥)})(𝐺𝑦) = (𝐺𝑥)))
3828, 35, 37syl2anc 691 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥) ↔ ∀𝑦 ∈ (𝐹 “ {(𝐹𝑥)})(𝐺𝑦) = (𝐺𝑥)))
3916, 38mpbird 246 . . . . . . . . . 10 ((𝜑𝑥𝑋) → ∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥))
40 fdm 5964 . . . . . . . . . . . . . . . 16 (𝐺:𝑋 𝐾 → dom 𝐺 = 𝑋)
4125, 40syl 17 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝐺 = 𝑋)
4241eleq2d 2673 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ dom 𝐺𝑥𝑋))
4342biimpar 501 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → 𝑥 ∈ dom 𝐺)
44 simpr 476 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → 𝑥𝑋)
45 eqidd 2611 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → (𝐹𝑥) = (𝐹𝑥))
46 fniniseg 6246 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝑋 → (𝑥 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑥𝑋 ∧ (𝐹𝑥) = (𝐹𝑥))))
474, 46syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → (𝑥 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑥𝑋 ∧ (𝐹𝑥) = (𝐹𝑥))))
4844, 45, 47mpbir2and 959 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → 𝑥 ∈ (𝐹 “ {(𝐹𝑥)}))
49 inelcm 3984 . . . . . . . . . . . . 13 ((𝑥 ∈ dom 𝐺𝑥 ∈ (𝐹 “ {(𝐹𝑥)})) → (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
5043, 48, 49syl2anc 691 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
51 imadisj 5403 . . . . . . . . . . . . 13 ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) = ∅ ↔ (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) = ∅)
5251necon3bii 2834 . . . . . . . . . . . 12 ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) ≠ ∅ ↔ (dom 𝐺 ∩ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
5350, 52sylibr 223 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ≠ ∅)
54 eqsn 4301 . . . . . . . . . . 11 ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) ≠ ∅ → ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)} ↔ ∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥)))
5553, 54syl 17 . . . . . . . . . 10 ((𝜑𝑥𝑋) → ((𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)} ↔ ∀𝑤 ∈ (𝐺 “ (𝐹 “ {(𝐹𝑥)}))𝑤 = (𝐺𝑥)))
5639, 55mpbird 246 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)})
5756unieqd 4382 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = {(𝐺𝑥)})
58 fvex 6113 . . . . . . . . 9 (𝐺𝑥) ∈ V
5958unisn 4387 . . . . . . . 8 {(𝐺𝑥)} = (𝐺𝑥)
6057, 59syl6req 2661 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐺𝑥) = (𝐺 “ (𝐹 “ {(𝐹𝑥)})))
6160mpteq2dva 4672 . . . . . 6 (𝜑 → (𝑥𝑋 ↦ (𝐺𝑥)) = (𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)}))))
6225feqmptd 6159 . . . . . 6 (𝜑𝐺 = (𝑥𝑋 ↦ (𝐺𝑥)))
6331ffvelrnda 6267 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ 𝑌)
6431feqmptd 6159 . . . . . . 7 (𝜑𝐹 = (𝑥𝑋 ↦ (𝐹𝑥)))
65 eqidd 2611 . . . . . . 7 (𝜑 → (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) = (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))))
66 sneq 4135 . . . . . . . . . 10 (𝑤 = (𝐹𝑥) → {𝑤} = {(𝐹𝑥)})
6766imaeq2d 5385 . . . . . . . . 9 (𝑤 = (𝐹𝑥) → (𝐹 “ {𝑤}) = (𝐹 “ {(𝐹𝑥)}))
6867imaeq2d 5385 . . . . . . . 8 (𝑤 = (𝐹𝑥) → (𝐺 “ (𝐹 “ {𝑤})) = (𝐺 “ (𝐹 “ {(𝐹𝑥)})))
6968unieqd 4382 . . . . . . 7 (𝑤 = (𝐹𝑥) → (𝐺 “ (𝐹 “ {𝑤})) = (𝐺 “ (𝐹 “ {(𝐹𝑥)})))
7063, 64, 65, 69fmptco 6303 . . . . . 6 (𝜑 → ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹) = (𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)}))))
7161, 62, 703eqtr4d 2654 . . . . 5 (𝜑𝐺 = ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹))
7271, 18eqeltrrd 2689 . . . 4 (𝜑 → ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹) ∈ (𝐽 Cn 𝐾))
7325ffvelrnda 6267 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝐺𝑥) ∈ 𝐾)
7460, 73eqeltrrd 2689 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾)
7574ralrimiva 2949 . . . . . . 7 (𝜑 → ∀𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾)
7669eqcomd 2616 . . . . . . . . . . 11 (𝑤 = (𝐹𝑥) → (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = (𝐺 “ (𝐹 “ {𝑤})))
7776eqcoms 2618 . . . . . . . . . 10 ((𝐹𝑥) = 𝑤 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) = (𝐺 “ (𝐹 “ {𝑤})))
7877eleq1d 2672 . . . . . . . . 9 ((𝐹𝑥) = 𝑤 → ( (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾))
7978cbvfo 6444 . . . . . . . 8 (𝐹:𝑋onto𝑌 → (∀𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾 ↔ ∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾))
801, 79syl 17 . . . . . . 7 (𝜑 → (∀𝑥𝑋 (𝐺 “ (𝐹 “ {(𝐹𝑥)})) ∈ 𝐾 ↔ ∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾))
8175, 80mpbid 221 . . . . . 6 (𝜑 → ∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾)
82 eqid 2610 . . . . . . 7 (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) = (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})))
8382fmpt 6289 . . . . . 6 (∀𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤})) ∈ 𝐾 ↔ (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))):𝑌 𝐾)
8481, 83sylib 207 . . . . 5 (𝜑 → (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))):𝑌 𝐾)
85 qtopcn 21327 . . . . 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)))
8617, 23, 1, 84, 85syl22anc 1319 . . . 4 (𝜑 → ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ↔ ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹) ∈ (𝐽 Cn 𝐾)))
8772, 86mpbird 246 . . 3 (𝜑 → (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
88 coeq1 5201 . . . . 5 (𝑓 = (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) → (𝑓𝐹) = ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹))
8988eqeq2d 2620 . . . 4 (𝑓 = (𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) → (𝐺 = (𝑓𝐹) ↔ 𝐺 = ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹)))
9089rspcev 3282 . . 3 (((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝐺 = ((𝑤𝑌 (𝐺 “ (𝐹 “ {𝑤}))) ∘ 𝐹)) → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹))
9187, 71, 90syl2anc 691 . 2 (𝜑 → ∃𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹))
92 eqtr2 2630 . . . 4 ((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → (𝑓𝐹) = (𝑔𝐹))
931adantr 480 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝐹:𝑋onto𝑌)
94 qtoptopon 21317 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋onto𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
9517, 1, 94syl2anc 691 . . . . . . . 8 (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
9695adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌))
9723adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝐾 ∈ (TopOn‘ 𝐾))
98 simprl 790 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
99 cnf2 20863 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)) → 𝑓:𝑌 𝐾)
10096, 97, 98, 99syl3anc 1318 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓:𝑌 𝐾)
101 ffn 5958 . . . . . 6 (𝑓:𝑌 𝐾𝑓 Fn 𝑌)
102100, 101syl 17 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑓 Fn 𝑌)
103 simprr 792 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))
104 cnf2 20863 . . . . . . 7 (((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)) → 𝑔:𝑌 𝐾)
10596, 97, 103, 104syl3anc 1318 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔:𝑌 𝐾)
106 ffn 5958 . . . . . 6 (𝑔:𝑌 𝐾𝑔 Fn 𝑌)
107105, 106syl 17 . . . . 5 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → 𝑔 Fn 𝑌)
108 cocan2 6447 . . . . 5 ((𝐹:𝑋onto𝑌𝑓 Fn 𝑌𝑔 Fn 𝑌) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
10993, 102, 107, 108syl3anc 1318 . . . 4 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → ((𝑓𝐹) = (𝑔𝐹) ↔ 𝑓 = 𝑔))
11092, 109syl5ib 233 . . 3 ((𝜑 ∧ (𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾) ∧ 𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾))) → ((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔))
111110ralrimivva 2954 . 2 (𝜑 → ∀𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)∀𝑔 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)((𝐺 = (𝑓𝐹) ∧ 𝐺 = (𝑔𝐹)) → 𝑓 = 𝑔))
112 coeq1 5201 . . . 4 (𝑓 = 𝑔 → (𝑓𝐹) = (𝑔𝐹))
113112eqeq2d 2620 . . 3 (𝑓 = 𝑔 → (𝐺 = (𝑓𝐹) ↔ 𝐺 = (𝑔𝐹)))
114113reu4 3367 . 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)))
11591, 111, 114sylanbrc 695 1 (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹) Cn 𝐾)𝐺 = (𝑓𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038   “ cima 5041   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   qTop cqtop 15986  Topctop 20517  TopOnctopon 20518   Cn ccn 20838 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 This theorem is referenced by:  qtophmeo  21430
 Copyright terms: Public domain W3C validator