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Theorem setcepi 17974
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 2736 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2736 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2736 . . . . . 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 17971 . . . . . . 7 (𝑈𝑉𝐶 ∈ Cat)
85, 7syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
9 setcmon.x . . . . . . 7 (𝜑𝑋𝑈)
106, 5setcbas 17964 . . . . . . 7 (𝜑𝑈 = (Base‘𝐶))
119, 10eleqtrd 2840 . . . . . 6 (𝜑𝑋 ∈ (Base‘𝐶))
12 setcmon.y . . . . . . 7 (𝜑𝑌𝑈)
1312, 10eleqtrd 2840 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐶))
141, 2, 3, 4, 8, 11, 13epihom 17625 . . . . 5 (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1514sselda 3944 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
166, 5, 2, 9, 12elsetchom 17967 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋𝑌))
1716biimpa 477 . . . 4 ((𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋𝑌)
1815, 17syldan 591 . . 3 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋𝑌)
1918frnd 6676 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹𝑌)
2018ffnd 6669 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 Fn 𝑋)
21 fnfvelrn 7031 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝑋𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2220, 21sylan 580 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2322iftrued 4494 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅) = 1o)
2423mpteq2dva 5205 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑥𝑋 ↦ if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅)) = (𝑥𝑋 ↦ 1o))
2518ffvelcdmda 7035 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ 𝑌)
2618feqmptd 6910 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 = (𝑥𝑋 ↦ (𝐹𝑥)))
27 eqidd 2737 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)))
28 eleq1 2825 . . . . . . . . . . . . 13 (𝑎 = (𝐹𝑥) → (𝑎 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
2928ifbid 4509 . . . . . . . . . . . 12 (𝑎 = (𝐹𝑥) → if(𝑎 ∈ ran 𝐹, 1o, ∅) = if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅))
3025, 26, 27, 29fmptco 7075 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = (𝑥𝑋 ↦ if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅)))
31 fconstmpt 5694 . . . . . . . . . . . . 13 (𝑌 × {1o}) = (𝑎𝑌 ↦ 1o)
3231a1i 11 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) = (𝑎𝑌 ↦ 1o))
33 eqidd 2737 . . . . . . . . . . . 12 (𝑎 = (𝐹𝑥) → 1o = 1o)
3425, 26, 32, 33fmptco 7075 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∘ 𝐹) = (𝑥𝑋 ↦ 1o))
3524, 30, 343eqtr4d 2786 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
365adantr 481 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑈𝑉)
379adantr 481 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋𝑈)
3812adantr 481 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌𝑈)
39 setcepi.2 . . . . . . . . . . . 12 (𝜑 → 2o𝑈)
4039adantr 481 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 2o𝑈)
41 eqid 2736 . . . . . . . . . . . . 13 (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))
42 1oex 8422 . . . . . . . . . . . . . . . . 17 1o ∈ V
4342prid2 4724 . . . . . . . . . . . . . . . 16 1o ∈ {∅, 1o}
44 df2o3 8420 . . . . . . . . . . . . . . . 16 2o = {∅, 1o}
4543, 44eleqtrri 2837 . . . . . . . . . . . . . . 15 1o ∈ 2o
46 0ex 5264 . . . . . . . . . . . . . . . . 17 ∅ ∈ V
4746prid1 4723 . . . . . . . . . . . . . . . 16 ∅ ∈ {∅, 1o}
4847, 44eleqtrri 2837 . . . . . . . . . . . . . . 15 ∅ ∈ 2o
4945, 48ifcli 4533 . . . . . . . . . . . . . 14 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
5049a1i 11 . . . . . . . . . . . . 13 (𝑎𝑌 → if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o)
5141, 50fmpti 7060 . . . . . . . . . . . 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 17969 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹))
54 fconst6g 6731 . . . . . . . . . . . 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 17969 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
5735, 53, 563eqtr4d 2786 . . . . . . . . 9 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹))
588adantr 481 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐶 ∈ Cat)
5911adantr 481 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ (Base‘𝐶))
6013adantr 481 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ (Base‘𝐶))
6139, 10eleqtrd 2840 . . . . . . . . . . 11 (𝜑 → 2o ∈ (Base‘𝐶))
6261adantr 481 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ (Base‘𝐶))
63 simpr 485 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋𝐸𝑌))
646, 36, 2, 38, 40elsetchom 17967 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o))
6552, 64mpbird 256 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o))
666, 36, 2, 38, 40elsetchom 17967 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑌 × {1o}):𝑌⟶2o))
6755, 66mpbird 256 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o))
681, 2, 3, 4, 58, 59, 60, 62, 63, 65, 67epii 17626 . . . . . . . . 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 231 . . . . . . . 8 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑌 × {1o}))
7069, 31eqtrdi 2792 . . . . . . 7 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ 1o))
7149rgenw 3068 . . . . . . . 8 𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
72 mpteqb 6967 . . . . . . . 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 217 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
75 1n0 8434 . . . . . . . . . 10 1o ≠ ∅
7675nesymi 3001 . . . . . . . . 9 ¬ ∅ = 1o
77 iffalse 4495 . . . . . . . . . 10 𝑎 ∈ ran 𝐹 → if(𝑎 ∈ ran 𝐹, 1o, ∅) = ∅)
7877eqeq1d 2738 . . . . . . . . 9 𝑎 ∈ ran 𝐹 → (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o ↔ ∅ = 1o))
7976, 78mtbiri 326 . . . . . . . 8 𝑎 ∈ ran 𝐹 → ¬ if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
8079con4i 114 . . . . . . 7 (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o𝑎 ∈ ran 𝐹)
8180ralimi 3086 . . . . . 6 (∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o → ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
8274, 81syl 17 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
83 dfss3 3932 . . . . 5 (𝑌 ⊆ ran 𝐹 ↔ ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
8482, 83sylibr 233 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ⊆ ran 𝐹)
8519, 84eqssd 3961 . . 3 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 = 𝑌)
86 dffo2 6760 . . 3 (𝐹:𝑋onto𝑌 ↔ (𝐹:𝑋𝑌 ∧ ran 𝐹 = 𝑌))
8718, 85, 86sylanbrc 583 . 2 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋onto𝑌)
88 fof 6756 . . . . 5 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
8988adantl 482 . . . 4 ((𝜑𝐹:𝑋onto𝑌) → 𝐹:𝑋𝑌)
9016biimpar 478 . . . 4 ((𝜑𝐹:𝑋𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
9189, 90syldan 591 . . 3 ((𝜑𝐹:𝑋onto𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
9210adantr 481 . . . . . 6 ((𝜑𝐹:𝑋onto𝑌) → 𝑈 = (Base‘𝐶))
9392eleq2d 2823 . . . . 5 ((𝜑𝐹:𝑋onto𝑌) → (𝑧𝑈𝑧 ∈ (Base‘𝐶)))
945ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
959ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑋𝑈)
9612ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑌𝑈)
97 simprl 769 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑧𝑈)
9889adantr 481 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋𝑌)
99 simprrl 779 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧))
1006, 94, 2, 96, 97elsetchom 17967 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ 𝑔:𝑌𝑧))
10199, 100mpbid 231 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔:𝑌𝑧)
1026, 94, 3, 95, 96, 97, 98, 101setcco 17969 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝑔𝐹))
103 simprrr 780 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ∈ (𝑌(Hom ‘𝐶)𝑧))
1046, 94, 2, 96, 97elsetchom 17967 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ( ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ :𝑌𝑧))
105103, 104mpbid 231 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → :𝑌𝑧)
1066, 94, 3, 95, 96, 97, 98, 105setcco 17969 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝐹))
107102, 106eqeq12d 2752 . . . . . . . . 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 767 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋onto𝑌)
109101ffnd 6669 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 Fn 𝑌)
110105ffnd 6669 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → Fn 𝑌)
111 cocan2 7238 . . . . . . . . . . 11 ((𝐹:𝑋onto𝑌𝑔 Fn 𝑌 Fn 𝑌) → ((𝑔𝐹) = (𝐹) ↔ 𝑔 = ))
112108, 109, 110, 111syl3anc 1371 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔𝐹) = (𝐹) ↔ 𝑔 = ))
113112biimpd 228 . . . . . . . . 9 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔𝐹) = (𝐹) → 𝑔 = ))
114107, 113sylbid 239 . . . . . . . 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 468 . . . . . . 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 3197 . . . . . 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 413 . . . . 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 259 . . . 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 3142 . . 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 17624 . . . 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 481 . . 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 711 . 2 ((𝜑𝐹:𝑋onto𝑌) → 𝐹 ∈ (𝑋𝐸𝑌))
12387, 122impbida 799 1 (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋onto𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wss 3910  c0 4282  ifcif 4486  {csn 4586  {cpr 4588  cop 4592  cmpt 5188   × cxp 5631  ran crn 5634  ccom 5637   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  1oc1o 8405  2oc2o 8406  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Epicepi 17612  SetCatcsetc 17961
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  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-nel 3050  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-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  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-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  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-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-oppc 17592  df-mon 17613  df-epi 17614  df-setc 17962
This theorem is referenced by: (None)
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