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Theorem setcepi 17339
Description: An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcmon.c 𝐶 = (SetCat‘𝑈)
setcmon.u (𝜑𝑈𝑉)
setcmon.x (𝜑𝑋𝑈)
setcmon.y (𝜑𝑌𝑈)
setcepi.h 𝐸 = (Epi‘𝐶)
setcepi.2 (𝜑 → 2o𝑈)
Assertion
Ref Expression
setcepi (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋onto𝑌))

Proof of Theorem setcepi
Dummy variables 𝑥 𝑔 𝑎 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2822 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2822 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2822 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
4 setcepi.h . . . . . 6 𝐸 = (Epi‘𝐶)
5 setcmon.u . . . . . . 7 (𝜑𝑈𝑉)
6 setcmon.c . . . . . . . 8 𝐶 = (SetCat‘𝑈)
76setccat 17336 . . . . . . 7 (𝑈𝑉𝐶 ∈ Cat)
85, 7syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
9 setcmon.x . . . . . . 7 (𝜑𝑋𝑈)
106, 5setcbas 17329 . . . . . . 7 (𝜑𝑈 = (Base‘𝐶))
119, 10eleqtrd 2916 . . . . . 6 (𝜑𝑋 ∈ (Base‘𝐶))
12 setcmon.y . . . . . . 7 (𝜑𝑌𝑈)
1312, 10eleqtrd 2916 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐶))
141, 2, 3, 4, 8, 11, 13epihom 17003 . . . . 5 (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1514sselda 3942 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
166, 5, 2, 9, 12elsetchom 17332 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋𝑌))
1716biimpa 480 . . . 4 ((𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋𝑌)
1815, 17syldan 594 . . 3 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋𝑌)
1918frnd 6501 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹𝑌)
2018ffnd 6495 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 Fn 𝑋)
21 fnfvelrn 6830 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝑋𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2220, 21sylan 583 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2322iftrued 4447 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅) = 1o)
2423mpteq2dva 5137 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑥𝑋 ↦ if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅)) = (𝑥𝑋 ↦ 1o))
2518ffvelrnda 6833 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ 𝑌)
2618feqmptd 6715 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 = (𝑥𝑋 ↦ (𝐹𝑥)))
27 eqidd 2823 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)))
28 eleq1 2901 . . . . . . . . . . . . 13 (𝑎 = (𝐹𝑥) → (𝑎 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
2928ifbid 4461 . . . . . . . . . . . 12 (𝑎 = (𝐹𝑥) → if(𝑎 ∈ ran 𝐹, 1o, ∅) = if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅))
3025, 26, 27, 29fmptco 6873 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = (𝑥𝑋 ↦ if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅)))
31 fconstmpt 5591 . . . . . . . . . . . . 13 (𝑌 × {1o}) = (𝑎𝑌 ↦ 1o)
3231a1i 11 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) = (𝑎𝑌 ↦ 1o))
33 eqidd 2823 . . . . . . . . . . . 12 (𝑎 = (𝐹𝑥) → 1o = 1o)
3425, 26, 32, 33fmptco 6873 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∘ 𝐹) = (𝑥𝑋 ↦ 1o))
3524, 30, 343eqtr4d 2867 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
365adantr 484 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑈𝑉)
379adantr 484 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋𝑈)
3812adantr 484 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌𝑈)
39 setcepi.2 . . . . . . . . . . . 12 (𝜑 → 2o𝑈)
4039adantr 484 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 2o𝑈)
41 eqid 2822 . . . . . . . . . . . . 13 (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))
42 1oex 8097 . . . . . . . . . . . . . . . . 17 1o ∈ V
4342prid2 4673 . . . . . . . . . . . . . . . 16 1o ∈ {∅, 1o}
44 df2o3 8104 . . . . . . . . . . . . . . . 16 2o = {∅, 1o}
4543, 44eleqtrri 2913 . . . . . . . . . . . . . . 15 1o ∈ 2o
46 0ex 5187 . . . . . . . . . . . . . . . . 17 ∅ ∈ V
4746prid1 4672 . . . . . . . . . . . . . . . 16 ∅ ∈ {∅, 1o}
4847, 44eleqtrri 2913 . . . . . . . . . . . . . . 15 ∅ ∈ 2o
4945, 48ifcli 4485 . . . . . . . . . . . . . 14 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
5049a1i 11 . . . . . . . . . . . . 13 (𝑎𝑌 → if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o)
5141, 50fmpti 6858 . . . . . . . . . . . 12 (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o
5251a1i 11 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o)
536, 36, 3, 37, 38, 40, 18, 52setcco 17334 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹))
54 fconst6g 6549 . . . . . . . . . . . 12 (1o ∈ 2o → (𝑌 × {1o}):𝑌⟶2o)
5545, 54mp1i 13 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}):𝑌⟶2o)
566, 36, 3, 37, 38, 40, 18, 55setcco 17334 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
5735, 53, 563eqtr4d 2867 . . . . . . . . 9 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹))
588adantr 484 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐶 ∈ Cat)
5911adantr 484 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ (Base‘𝐶))
6013adantr 484 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ (Base‘𝐶))
6139, 10eleqtrd 2916 . . . . . . . . . . 11 (𝜑 → 2o ∈ (Base‘𝐶))
6261adantr 484 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ (Base‘𝐶))
63 simpr 488 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋𝐸𝑌))
646, 36, 2, 38, 40elsetchom 17332 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o))
6552, 64mpbird 260 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o))
666, 36, 2, 38, 40elsetchom 17332 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑌 × {1o}):𝑌⟶2o))
6755, 66mpbird 260 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o))
681, 2, 3, 4, 58, 59, 60, 62, 63, 65, 67epii 17004 . . . . . . . . 9 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) ↔ (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑌 × {1o})))
6957, 68mpbid 235 . . . . . . . 8 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑌 × {1o}))
7069, 31syl6eq 2873 . . . . . . 7 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ 1o))
7149rgenw 3142 . . . . . . . 8 𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
72 mpteqb 6769 . . . . . . . 8 (∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ 1o) ↔ ∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o))
7371, 72ax-mp 5 . . . . . . 7 ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ 1o) ↔ ∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
7470, 73sylib 221 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
75 1n0 8106 . . . . . . . . . 10 1o ≠ ∅
7675nesymi 3068 . . . . . . . . 9 ¬ ∅ = 1o
77 iffalse 4448 . . . . . . . . . 10 𝑎 ∈ ran 𝐹 → if(𝑎 ∈ ran 𝐹, 1o, ∅) = ∅)
7877eqeq1d 2824 . . . . . . . . 9 𝑎 ∈ ran 𝐹 → (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o ↔ ∅ = 1o))
7976, 78mtbiri 330 . . . . . . . 8 𝑎 ∈ ran 𝐹 → ¬ if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
8079con4i 114 . . . . . . 7 (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o𝑎 ∈ ran 𝐹)
8180ralimi 3152 . . . . . 6 (∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o → ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
8274, 81syl 17 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
83 dfss3 3930 . . . . 5 (𝑌 ⊆ ran 𝐹 ↔ ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
8482, 83sylibr 237 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ⊆ ran 𝐹)
8519, 84eqssd 3959 . . 3 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 = 𝑌)
86 dffo2 6576 . . 3 (𝐹:𝑋onto𝑌 ↔ (𝐹:𝑋𝑌 ∧ ran 𝐹 = 𝑌))
8718, 85, 86sylanbrc 586 . 2 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋onto𝑌)
88 fof 6572 . . . . 5 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
8988adantl 485 . . . 4 ((𝜑𝐹:𝑋onto𝑌) → 𝐹:𝑋𝑌)
9016biimpar 481 . . . 4 ((𝜑𝐹:𝑋𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
9189, 90syldan 594 . . 3 ((𝜑𝐹:𝑋onto𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
9210adantr 484 . . . . . 6 ((𝜑𝐹:𝑋onto𝑌) → 𝑈 = (Base‘𝐶))
9392eleq2d 2899 . . . . 5 ((𝜑𝐹:𝑋onto𝑌) → (𝑧𝑈𝑧 ∈ (Base‘𝐶)))
945ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
959ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑋𝑈)
9612ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑌𝑈)
97 simprl 770 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑧𝑈)
9889adantr 484 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋𝑌)
99 simprrl 780 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧))
1006, 94, 2, 96, 97elsetchom 17332 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ 𝑔:𝑌𝑧))
10199, 100mpbid 235 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔:𝑌𝑧)
1026, 94, 3, 95, 96, 97, 98, 101setcco 17334 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝑔𝐹))
103 simprrr 781 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ∈ (𝑌(Hom ‘𝐶)𝑧))
1046, 94, 2, 96, 97elsetchom 17332 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ( ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ :𝑌𝑧))
105103, 104mpbid 235 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → :𝑌𝑧)
1066, 94, 3, 95, 96, 97, 98, 105setcco 17334 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝐹))
107102, 106eqeq12d 2838 . . . . . . . . 9 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) ↔ (𝑔𝐹) = (𝐹)))
108 simplr 768 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋onto𝑌)
109101ffnd 6495 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 Fn 𝑌)
110105ffnd 6495 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → Fn 𝑌)
111 cocan2 7031 . . . . . . . . . . 11 ((𝐹:𝑋onto𝑌𝑔 Fn 𝑌 Fn 𝑌) → ((𝑔𝐹) = (𝐹) ↔ 𝑔 = ))
112108, 109, 110, 111syl3anc 1368 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔𝐹) = (𝐹) ↔ 𝑔 = ))
113112biimpd 232 . . . . . . . . 9 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔𝐹) = (𝐹) → 𝑔 = ))
114107, 113sylbid 243 . . . . . . . 8 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))
115114anassrs 471 . . . . . . 7 ((((𝜑𝐹:𝑋onto𝑌) ∧ 𝑧𝑈) ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))
116115ralrimivva 3181 . . . . . 6 (((𝜑𝐹:𝑋onto𝑌) ∧ 𝑧𝑈) → ∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))
117116ex 416 . . . . 5 ((𝜑𝐹:𝑋onto𝑌) → (𝑧𝑈 → ∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = )))
11893, 117sylbird 263 . . . 4 ((𝜑𝐹:𝑋onto𝑌) → (𝑧 ∈ (Base‘𝐶) → ∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = )))
119118ralrimiv 3173 . . 3 ((𝜑𝐹:𝑋onto𝑌) → ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))
1201, 2, 3, 4, 8, 11, 13isepi2 17002 . . . 4 (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))))
121120adantr 484 . . 3 ((𝜑𝐹:𝑋onto𝑌) → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ))))
12291, 119, 121mpbir2and 712 . 2 ((𝜑𝐹:𝑋onto𝑌) → 𝐹 ∈ (𝑋𝐸𝑌))
12387, 122impbida 800 1 (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋onto𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wral 3130  wss 3908  c0 4265  ifcif 4439  {csn 4539  {cpr 4541  cop 4545  cmpt 5122   × cxp 5530  ran crn 5533  ccom 5536   Fn wfn 6329  wf 6330  ontowfo 6332  cfv 6334  (class class class)co 7140  1oc1o 8082  2oc2o 8083  Basecbs 16474  Hom chom 16567  compcco 16568  Catccat 16926  Epicepi 16990  SetCatcsetc 17326
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-struct 16476  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-hom 16580  df-cco 16581  df-cat 16930  df-cid 16931  df-oppc 16973  df-mon 16991  df-epi 16992  df-setc 17327
This theorem is referenced by: (None)
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