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

Theorem setcepi 17200
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 2775 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2775 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2775 . . . . . 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 17197 . . . . . . 7 (𝑈𝑉𝐶 ∈ Cat)
85, 7syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
9 setcmon.x . . . . . . 7 (𝜑𝑋𝑈)
106, 5setcbas 17190 . . . . . . 7 (𝜑𝑈 = (Base‘𝐶))
119, 10eleqtrd 2865 . . . . . 6 (𝜑𝑋 ∈ (Base‘𝐶))
12 setcmon.y . . . . . . 7 (𝜑𝑌𝑈)
1312, 10eleqtrd 2865 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐶))
141, 2, 3, 4, 8, 11, 13epihom 16864 . . . . 5 (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1514sselda 3857 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
166, 5, 2, 9, 12elsetchom 17193 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋𝑌))
1716biimpa 469 . . . 4 ((𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋𝑌)
1815, 17syldan 582 . . 3 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋𝑌)
1918frnd 6349 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹𝑌)
2018ffnd 6343 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 Fn 𝑋)
21 fnfvelrn 6671 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝑋𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2220, 21sylan 572 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ ran 𝐹)
2322iftrued 4356 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅) = 1o)
2423mpteq2dva 5020 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑥𝑋 ↦ if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅)) = (𝑥𝑋 ↦ 1o))
2518ffvelrnda 6674 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ 𝑌)
2618feqmptd 6560 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 = (𝑥𝑋 ↦ (𝐹𝑥)))
27 eqidd 2776 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)))
28 eleq1 2850 . . . . . . . . . . . . 13 (𝑎 = (𝐹𝑥) → (𝑎 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
2928ifbid 4370 . . . . . . . . . . . 12 (𝑎 = (𝐹𝑥) → if(𝑎 ∈ ran 𝐹, 1o, ∅) = if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅))
3025, 26, 27, 29fmptco 6712 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = (𝑥𝑋 ↦ if((𝐹𝑥) ∈ ran 𝐹, 1o, ∅)))
31 fconstmpt 5461 . . . . . . . . . . . . 13 (𝑌 × {1o}) = (𝑎𝑌 ↦ 1o)
3231a1i 11 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) = (𝑎𝑌 ↦ 1o))
33 eqidd 2776 . . . . . . . . . . . 12 (𝑎 = (𝐹𝑥) → 1o = 1o)
3425, 26, 32, 33fmptco 6712 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∘ 𝐹) = (𝑥𝑋 ↦ 1o))
3524, 30, 343eqtr4d 2821 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
365adantr 473 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑈𝑉)
379adantr 473 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋𝑈)
3812adantr 473 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌𝑈)
39 setcepi.2 . . . . . . . . . . . 12 (𝜑 → 2o𝑈)
4039adantr 473 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 2o𝑈)
41 eqid 2775 . . . . . . . . . . . . 13 (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))
42 1oex 7909 . . . . . . . . . . . . . . . . 17 1o ∈ V
4342prid2 4571 . . . . . . . . . . . . . . . 16 1o ∈ {∅, 1o}
44 df2o3 7915 . . . . . . . . . . . . . . . 16 2o = {∅, 1o}
4543, 44eleqtrri 2862 . . . . . . . . . . . . . . 15 1o ∈ 2o
46 0ex 5066 . . . . . . . . . . . . . . . . 17 ∅ ∈ V
4746prid1 4570 . . . . . . . . . . . . . . . 16 ∅ ∈ {∅, 1o}
4847, 44eleqtrri 2862 . . . . . . . . . . . . . . 15 ∅ ∈ 2o
4945, 48ifcli 4394 . . . . . . . . . . . . . 14 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
5049a1i 11 . . . . . . . . . . . . 13 (𝑎𝑌 → if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o)
5141, 50fmpti 6697 . . . . . . . . . . . 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 17195 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹))
54 fconst6g 6395 . . . . . . . . . . . 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 17195 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
5735, 53, 563eqtr4d 2821 . . . . . . . . 9 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹))
588adantr 473 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐶 ∈ Cat)
5911adantr 473 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ (Base‘𝐶))
6013adantr 473 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ (Base‘𝐶))
6139, 10eleqtrd 2865 . . . . . . . . . . 11 (𝜑 → 2o ∈ (Base‘𝐶))
6261adantr 473 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ (Base‘𝐶))
63 simpr 477 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋𝐸𝑌))
646, 36, 2, 38, 40elsetchom 17193 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o))
6552, 64mpbird 249 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o))
666, 36, 2, 38, 40elsetchom 17193 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑌 × {1o}):𝑌⟶2o))
6755, 66mpbird 249 . . . . . . . . . 10 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o))
681, 2, 3, 4, 58, 59, 60, 62, 63, 65, 67epii 16865 . . . . . . . . 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 224 . . . . . . . 8 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑌 × {1o}))
7069, 31syl6eq 2827 . . . . . . 7 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎𝑌 ↦ 1o))
7149rgenw 3097 . . . . . . . 8 𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
72 mpteqb 6611 . . . . . . . 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 210 . . . . . 6 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
75 1n0 7917 . . . . . . . . . 10 1o ≠ ∅
7675nesymi 3021 . . . . . . . . 9 ¬ ∅ = 1o
77 iffalse 4357 . . . . . . . . . 10 𝑎 ∈ ran 𝐹 → if(𝑎 ∈ ran 𝐹, 1o, ∅) = ∅)
7877eqeq1d 2777 . . . . . . . . 9 𝑎 ∈ ran 𝐹 → (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o ↔ ∅ = 1o))
7976, 78mtbiri 319 . . . . . . . 8 𝑎 ∈ ran 𝐹 → ¬ if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
8079con4i 114 . . . . . . 7 (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o𝑎 ∈ ran 𝐹)
8180ralimi 3107 . . . . . 6 (∀𝑎𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o → ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
8274, 81syl 17 . . . . 5 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
83 dfss3 3846 . . . . 5 (𝑌 ⊆ ran 𝐹 ↔ ∀𝑎𝑌 𝑎 ∈ ran 𝐹)
8482, 83sylibr 226 . . . 4 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ⊆ ran 𝐹)
8519, 84eqssd 3874 . . 3 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 = 𝑌)
86 dffo2 6421 . . 3 (𝐹:𝑋onto𝑌 ↔ (𝐹:𝑋𝑌 ∧ ran 𝐹 = 𝑌))
8718, 85, 86sylanbrc 575 . 2 ((𝜑𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋onto𝑌)
88 fof 6417 . . . . 5 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
8988adantl 474 . . . 4 ((𝜑𝐹:𝑋onto𝑌) → 𝐹:𝑋𝑌)
9016biimpar 470 . . . 4 ((𝜑𝐹:𝑋𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
9189, 90syldan 582 . . 3 ((𝜑𝐹:𝑋onto𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
9210adantr 473 . . . . . 6 ((𝜑𝐹:𝑋onto𝑌) → 𝑈 = (Base‘𝐶))
9392eleq2d 2848 . . . . 5 ((𝜑𝐹:𝑋onto𝑌) → (𝑧𝑈𝑧 ∈ (Base‘𝐶)))
945ad2antrr 713 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
959ad2antrr 713 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑋𝑈)
9612ad2antrr 713 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑌𝑈)
97 simprl 758 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑧𝑈)
9889adantr 473 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋𝑌)
99 simprrl 768 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧))
1006, 94, 2, 96, 97elsetchom 17193 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ 𝑔:𝑌𝑧))
10199, 100mpbid 224 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔:𝑌𝑧)
1026, 94, 3, 95, 96, 97, 98, 101setcco 17195 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝑔𝐹))
103 simprrr 769 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ∈ (𝑌(Hom ‘𝐶)𝑧))
1046, 94, 2, 96, 97elsetchom 17193 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ( ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ :𝑌𝑧))
105103, 104mpbid 224 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → :𝑌𝑧)
1066, 94, 3, 95, 96, 97, 98, 105setcco 17195 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝐹))
107102, 106eqeq12d 2790 . . . . . . . . 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 756 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋onto𝑌)
109101ffnd 6343 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 Fn 𝑌)
110105ffnd 6343 . . . . . . . . . . 11 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → Fn 𝑌)
111 cocan2 6871 . . . . . . . . . . 11 ((𝐹:𝑋onto𝑌𝑔 Fn 𝑌 Fn 𝑌) → ((𝑔𝐹) = (𝐹) ↔ 𝑔 = ))
112108, 109, 110, 111syl3anc 1351 . . . . . . . . . 10 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔𝐹) = (𝐹) ↔ 𝑔 = ))
113112biimpd 221 . . . . . . . . 9 (((𝜑𝐹:𝑋onto𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔𝐹) = (𝐹) → 𝑔 = ))
114107, 113sylbid 232 . . . . . . . 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 460 . . . . . . 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 3138 . . . . . 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 405 . . . . 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 252 . . . 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 3128 . . 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 16863 . . . 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 473 . . 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 700 . 2 ((𝜑𝐹:𝑋onto𝑌) → 𝐹 ∈ (𝑋𝐸𝑌))
12387, 122impbida 788 1 (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋onto𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387   = wceq 1507  wcel 2048  wral 3085  wss 3828  c0 4177  ifcif 4348  {csn 4439  {cpr 4441  cop 4445  cmpt 5006   × cxp 5402  ran crn 5405  ccom 5408   Fn wfn 6181  wf 6182  ontowfo 6184  cfv 6186  (class class class)co 6974  1oc1o 7894  2oc2o 7895  Basecbs 16333  Hom chom 16426  compcco 16427  Catccat 16787  Epicepi 16851  SetCatcsetc 17187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-rep 5047  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277  ax-cnex 10387  ax-resscn 10388  ax-1cn 10389  ax-icn 10390  ax-addcl 10391  ax-addrcl 10392  ax-mulcl 10393  ax-mulrcl 10394  ax-mulcom 10395  ax-addass 10396  ax-mulass 10397  ax-distr 10398  ax-i2m1 10399  ax-1ne0 10400  ax-1rid 10401  ax-rnegex 10402  ax-rrecex 10403  ax-cnre 10404  ax-pre-lttri 10405  ax-pre-lttrn 10406  ax-pre-ltadd 10407  ax-pre-mulgt0 10408
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ne 2965  df-nel 3071  df-ral 3090  df-rex 3091  df-reu 3092  df-rmo 3093  df-rab 3094  df-v 3414  df-sbc 3681  df-csb 3786  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-pss 3844  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-tp 4444  df-op 4446  df-uni 4711  df-int 4748  df-iun 4792  df-br 4928  df-opab 4990  df-mpt 5007  df-tr 5029  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7498  df-2nd 7499  df-tpos 7692  df-wrecs 7747  df-recs 7809  df-rdg 7847  df-1o 7901  df-2o 7902  df-oadd 7905  df-er 8085  df-map 8204  df-en 8303  df-dom 8304  df-sdom 8305  df-fin 8306  df-pnf 10472  df-mnf 10473  df-xr 10474  df-ltxr 10475  df-le 10476  df-sub 10668  df-neg 10669  df-nn 11436  df-2 11500  df-3 11501  df-4 11502  df-5 11503  df-6 11504  df-7 11505  df-8 11506  df-9 11507  df-n0 11705  df-z 11791  df-dec 11909  df-uz 12056  df-fz 12706  df-struct 16335  df-ndx 16336  df-slot 16337  df-base 16339  df-sets 16340  df-hom 16439  df-cco 16440  df-cat 16791  df-cid 16792  df-oppc 16834  df-mon 16852  df-epi 16853  df-setc 17188
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator