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Theorem setcepi 18141
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 2734 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2734 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2734 . . . . . 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 18138 . . . . . . 7 (𝑈𝑉𝐶 ∈ Cat)
85, 7syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
9 setcmon.x . . . . . . 7 (𝜑𝑋𝑈)
106, 5setcbas 18131 . . . . . . 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 17789 . . . . 5 (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1514sselda 3994 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
166, 5, 2, 9, 12elsetchom 18134 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋𝑌))
1716biimpa 476 . . . 4 ((𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋𝑌)
1815, 17syldan 591 . . 3 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋𝑌)
1918frnd 6744 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹𝑌)
2018ffnd 6737 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 Fn 𝑋)
21 fnfvelrn 7099 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝑋𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2220, 21sylan 580 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2322iftrued 4538 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅) = 1o)
2423mpteq2dva 5247 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑥𝑋 ↦ if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅)) = (𝑥𝑋 ↦ 1o))
2518ffvelcdmda 7103 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ 𝑌)
2618feqmptd 6976 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 = (𝑥𝑋 ↦ (𝐹𝑥)))
27 eqidd 2735 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)))
28 eleq1 2826 . . . . . . . . . . . . 13 (𝑎 = (𝐹𝑥) → (𝑎 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
2928ifbid 4553 . . . . . . . . . . . 12 (𝑎 = (𝐹𝑥) → if(𝑎 ∈ ran 𝐹, 1o, ∅) = if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅))
3025, 26, 27, 29fmptco 7148 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = (𝑥𝑋 ↦ if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅)))
31 fconstmpt 5750 . . . . . . . . . . . . 13 (𝑌 × {1o}) = (𝑎𝑌 ↦ 1o)
3231a1i 11 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) = (𝑎𝑌 ↦ 1o))
33 eqidd 2735 . . . . . . . . . . . 12 (𝑎 = (𝐹𝑥) → 1o = 1o)
3425, 26, 32, 33fmptco 7148 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∘ 𝐹) = (𝑥𝑋 ↦ 1o))
3524, 30, 343eqtr4d 2784 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
365adantr 480 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑈𝑉)
379adantr 480 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋𝑈)
3812adantr 480 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌𝑈)
39 setcepi.2 . . . . . . . . . . . 12 (𝜑 → 2o𝑈)
4039adantr 480 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 2o𝑈)
41 eqid 2734 . . . . . . . . . . . . 13 (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))
42 1oex 8514 . . . . . . . . . . . . . . . . 17 1o ∈ V
4342prid2 4767 . . . . . . . . . . . . . . . 16 1o ∈ {∅, 1o}
44 df2o3 8512 . . . . . . . . . . . . . . . 16 2o = {∅, 1o}
4543, 44eleqtrri 2837 . . . . . . . . . . . . . . 15 1o ∈ 2o
46 0ex 5312 . . . . . . . . . . . . . . . . 17 ∅ ∈ V
4746prid1 4766 . . . . . . . . . . . . . . . 16 ∅ ∈ {∅, 1o}
4847, 44eleqtrri 2837 . . . . . . . . . . . . . . 15 ∅ ∈ 2o
4945, 48ifcli 4577 . . . . . . . . . . . . . 14 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
5049a1i 11 . . . . . . . . . . . . 13 (𝑎𝑌 → if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o)
5141, 50fmpti 7131 . . . . . . . . . . . 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 18136 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹))
54 fconst6g 6797 . . . . . . . . . . . 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 18136 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
5735, 53, 563eqtr4d 2784 . . . . . . . . 9 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹))
588adantr 480 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐶 ∈ Cat)
5911adantr 480 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ (Base‘𝐶))
6013adantr 480 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ (Base‘𝐶))
6139, 10eleqtrd 2840 . . . . . . . . . . 11 (𝜑 → 2o ∈ (Base‘𝐶))
6261adantr 480 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ (Base‘𝐶))
63 simpr 484 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋𝐸𝑌))
646, 36, 2, 38, 40elsetchom 18134 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o))
6552, 64mpbird 257 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o))
666, 36, 2, 38, 40elsetchom 18134 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑌 × {1o}):𝑌⟶2o))
6755, 66mpbird 257 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o))
681, 2, 3, 4, 58, 59, 60, 62, 63, 65, 67epii 17790 . . . . . . . . 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 232 . . . . . . . 8 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑌 × {1o}))
7069, 31eqtrdi 2790 . . . . . . 7 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ 1o))
7149rgenw 3062 . . . . . . . 8 𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
72 mpteqb 7034 . . . . . . . 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 218 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
75 1n0 8524 . . . . . . . . . 10 1o ≠ ∅
7675nesymi 2995 . . . . . . . . 9 ¬ ∅ = 1o
77 iffalse 4539 . . . . . . . . . 10 𝑎 ∈ ran 𝐹 → if(𝑎 ∈ ran 𝐹, 1o, ∅) = ∅)
7877eqeq1d 2736 . . . . . . . . 9 𝑎 ∈ ran 𝐹 → (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o ↔ ∅ = 1o))
7976, 78mtbiri 327 . . . . . . . 8 𝑎 ∈ ran 𝐹 → ¬ if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
8079con4i 114 . . . . . . 7 (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o𝑎 ∈ ran 𝐹)
8180ralimi 3080 . . . . . 6 (∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o → ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
8274, 81syl 17 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
83 dfss3 3983 . . . . 5 (𝑌 ⊆ ran 𝐹 ↔ ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
8482, 83sylibr 234 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ⊆ ran 𝐹)
8519, 84eqssd 4012 . . 3 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 = 𝑌)
86 dffo2 6824 . . 3 (𝐹:𝑋onto𝑌 ↔ (𝐹:𝑋𝑌 ∧ ran 𝐹 = 𝑌))
8718, 85, 86sylanbrc 583 . 2 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋onto𝑌)
88 fof 6820 . . . . 5 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
8988adantl 481 . . . 4 ((𝜑𝐹:𝑋onto𝑌) → 𝐹:𝑋𝑌)
9016biimpar 477 . . . 4 ((𝜑𝐹:𝑋𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
9189, 90syldan 591 . . 3 ((𝜑𝐹:𝑋onto𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
9210adantr 480 . . . . . 6 ((𝜑𝐹:𝑋onto𝑌) → 𝑈 = (Base‘𝐶))
9392eleq2d 2824 . . . . 5 ((𝜑𝐹:𝑋onto𝑌) → (𝑧𝑈𝑧 ∈ (Base‘𝐶)))
945ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
959ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑋𝑈)
9612ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑌𝑈)
97 simprl 771 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑧𝑈)
9889adantr 480 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋𝑌)
99 simprrl 781 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧))
1006, 94, 2, 96, 97elsetchom 18134 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ 𝑔:𝑌𝑧))
10199, 100mpbid 232 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔:𝑌𝑧)
1026, 94, 3, 95, 96, 97, 98, 101setcco 18136 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝑔𝐹))
103 simprrr 782 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ∈ (𝑌(Hom ‘𝐶)𝑧))
1046, 94, 2, 96, 97elsetchom 18134 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ( ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ :𝑌𝑧))
105103, 104mpbid 232 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → :𝑌𝑧)
1066, 94, 3, 95, 96, 97, 98, 105setcco 18136 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝐹))
107102, 106eqeq12d 2750 . . . . . . . . 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 769 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋onto𝑌)
109101ffnd 6737 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 Fn 𝑌)
110105ffnd 6737 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → Fn 𝑌)
111 cocan2 7311 . . . . . . . . . . 11 ((𝐹:𝑋onto𝑌𝑔 Fn 𝑌 Fn 𝑌) → ((𝑔𝐹) = (𝐹) ↔ 𝑔 = ))
112108, 109, 110, 111syl3anc 1370 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔𝐹) = (𝐹) ↔ 𝑔 = ))
113112biimpd 229 . . . . . . . . 9 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔𝐹) = (𝐹) → 𝑔 = ))
114107, 113sylbid 240 . . . . . . . 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 467 . . . . . . 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 3199 . . . . . 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 412 . . . . 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 260 . . . 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 17788 . . . 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 480 . . 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 713 . 2 ((𝜑𝐹:𝑋onto𝑌) → 𝐹 ∈ (𝑋𝐸𝑌))
12387, 122impbida 801 1 (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋onto𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wss 3962  c0 4338  ifcif 4530  {csn 4630  {cpr 4632  cop 4636  cmpt 5230   × cxp 5686  ran crn 5689  ccom 5692   Fn wfn 6557  wf 6558  ontowfo 6560  cfv 6562  (class class class)co 7430  1oc1o 8497  2oc2o 8498  Basecbs 17244  Hom chom 17308  compcco 17309  Catccat 17708  Epicepi 17776  SetCatcsetc 18128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-hom 17321  df-cco 17322  df-cat 17712  df-cid 17713  df-oppc 17756  df-mon 17777  df-epi 17778  df-setc 18129
This theorem is referenced by: (None)
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