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Theorem setcmon 16859
Description: A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcmon.c 𝐶 = (SetCat‘𝑈)
setcmon.u (𝜑𝑈𝑉)
setcmon.x (𝜑𝑋𝑈)
setcmon.y (𝜑𝑌𝑈)
setcmon.h 𝑀 = (Mono‘𝐶)
Assertion
Ref Expression
setcmon (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋1-1𝑌))

Proof of Theorem setcmon
Dummy variables 𝑥 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2724 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2724 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2724 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
4 setcmon.h . . . . . 6 𝑀 = (Mono‘𝐶)
5 setcmon.u . . . . . . 7 (𝜑𝑈𝑉)
6 setcmon.c . . . . . . . 8 𝐶 = (SetCat‘𝑈)
76setccat 16857 . . . . . . 7 (𝑈𝑉𝐶 ∈ Cat)
85, 7syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
9 setcmon.x . . . . . . 7 (𝜑𝑋𝑈)
106, 5setcbas 16850 . . . . . . 7 (𝜑𝑈 = (Base‘𝐶))
119, 10eleqtrd 2805 . . . . . 6 (𝜑𝑋 ∈ (Base‘𝐶))
12 setcmon.y . . . . . . 7 (𝜑𝑌𝑈)
1312, 10eleqtrd 2805 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐶))
141, 2, 3, 4, 8, 11, 13monhom 16517 . . . . 5 (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1514sselda 3709 . . . 4 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
166, 5, 2, 9, 12elsetchom 16853 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋𝑌))
1716biimpa 502 . . . 4 ((𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋𝑌)
1815, 17syldan 488 . . 3 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋𝑌)
19 simprr 813 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
2019sneqd 4297 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → {(𝐹𝑥)} = {(𝐹𝑦)})
2120xpeq2d 5248 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {(𝐹𝑥)}) = (𝑋 × {(𝐹𝑦)}))
2218adantr 472 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹:𝑋𝑌)
23 ffn 6158 . . . . . . . . . . . 12 (𝐹:𝑋𝑌𝐹 Fn 𝑋)
2422, 23syl 17 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹 Fn 𝑋)
25 simprll 821 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥𝑋)
26 fcoconst 6516 . . . . . . . . . . 11 ((𝐹 Fn 𝑋𝑥𝑋) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹𝑥)}))
2724, 25, 26syl2anc 696 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹𝑥)}))
28 simprlr 822 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑦𝑋)
29 fcoconst 6516 . . . . . . . . . . 11 ((𝐹 Fn 𝑋𝑦𝑋) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹𝑦)}))
3024, 28, 29syl2anc 696 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹𝑦)}))
3121, 27, 303eqtr4d 2768 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑦})))
325ad2antrr 764 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑈𝑉)
339ad2antrr 764 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑋𝑈)
3412ad2antrr 764 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑌𝑈)
35 fconst6g 6207 . . . . . . . . . . 11 (𝑥𝑋 → (𝑋 × {𝑥}):𝑋𝑋)
3625, 35syl 17 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑥}):𝑋𝑋)
376, 32, 3, 33, 33, 34, 36, 22setcco 16855 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑥})))
38 fconst6g 6207 . . . . . . . . . . 11 (𝑦𝑋 → (𝑋 × {𝑦}):𝑋𝑋)
3928, 38syl 17 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑦}):𝑋𝑋)
406, 32, 3, 33, 33, 34, 39, 22setcco 16855 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})) = (𝐹 ∘ (𝑋 × {𝑦})))
4131, 37, 403eqtr4d 2768 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})))
428ad2antrr 764 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐶 ∈ Cat)
4311ad2antrr 764 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑋 ∈ (Base‘𝐶))
4413ad2antrr 764 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑌 ∈ (Base‘𝐶))
45 simplr 809 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹 ∈ (𝑋𝑀𝑌))
466, 32, 2, 33, 33elsetchom 16853 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥}) ∈ (𝑋(Hom ‘𝐶)𝑋) ↔ (𝑋 × {𝑥}):𝑋𝑋))
4736, 46mpbird 247 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑥}) ∈ (𝑋(Hom ‘𝐶)𝑋))
486, 32, 2, 33, 33elsetchom 16853 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑦}) ∈ (𝑋(Hom ‘𝐶)𝑋) ↔ (𝑋 × {𝑦}):𝑋𝑋))
4939, 48mpbird 247 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑦}) ∈ (𝑋(Hom ‘𝐶)𝑋))
501, 2, 3, 4, 42, 43, 44, 43, 45, 47, 49moni 16518 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})) ↔ (𝑋 × {𝑥}) = (𝑋 × {𝑦})))
5141, 50mpbid 222 . . . . . . 7 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑥}) = (𝑋 × {𝑦}))
5251fveq1d 6306 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥})‘𝑥) = ((𝑋 × {𝑦})‘𝑥))
53 vex 3307 . . . . . . . 8 𝑥 ∈ V
5453fvconst2 6585 . . . . . . 7 (𝑥𝑋 → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
5525, 54syl 17 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
56 vex 3307 . . . . . . . 8 𝑦 ∈ V
5756fvconst2 6585 . . . . . . 7 (𝑥𝑋 → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
5825, 57syl 17 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
5952, 55, 583eqtr3d 2766 . . . . 5 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥 = 𝑦)
6059expr 644 . . . 4 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6160ralrimivva 3073 . . 3 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
62 dff13 6627 . . 3 (𝐹:𝑋1-1𝑌 ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6318, 61, 62sylanbrc 701 . 2 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋1-1𝑌)
64 f1f 6214 . . . 4 (𝐹:𝑋1-1𝑌𝐹:𝑋𝑌)
6516biimpar 503 . . . 4 ((𝜑𝐹:𝑋𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
6664, 65sylan2 492 . . 3 ((𝜑𝐹:𝑋1-1𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
6710adantr 472 . . . . . 6 ((𝜑𝐹:𝑋1-1𝑌) → 𝑈 = (Base‘𝐶))
6867eleq2d 2789 . . . . 5 ((𝜑𝐹:𝑋1-1𝑌) → (𝑧𝑈𝑧 ∈ (Base‘𝐶)))
695ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑈𝑉)
70 simprl 811 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑧𝑈)
719ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑋𝑈)
7212ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑌𝑈)
73 simprrl 823 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋))
746, 69, 2, 70, 71elsetchom 16853 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↔ 𝑔:𝑧𝑋))
7573, 74mpbid 222 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑔:𝑧𝑋)
7664ad2antlr 765 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝐹:𝑋𝑌)
776, 69, 3, 70, 71, 72, 75, 76setcco 16855 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹𝑔))
78 simprrr 824 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ∈ (𝑧(Hom ‘𝐶)𝑋))
796, 69, 2, 70, 71elsetchom 16853 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ( ∈ (𝑧(Hom ‘𝐶)𝑋) ↔ :𝑧𝑋))
8078, 79mpbid 222 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → :𝑧𝑋)
816, 69, 3, 70, 71, 72, 80, 76setcco 16855 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) = (𝐹))
8277, 81eqeq12d 2739 . . . . . . . . 9 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) ↔ (𝐹𝑔) = (𝐹)))
83 simplr 809 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝐹:𝑋1-1𝑌)
84 cocan1 6661 . . . . . . . . . . 11 ((𝐹:𝑋1-1𝑌𝑔:𝑧𝑋:𝑧𝑋) → ((𝐹𝑔) = (𝐹) ↔ 𝑔 = ))
8583, 75, 80, 84syl3anc 1439 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹𝑔) = (𝐹) ↔ 𝑔 = ))
8685biimpd 219 . . . . . . . . 9 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹𝑔) = (𝐹) → 𝑔 = ))
8782, 86sylbid 230 . . . . . . . 8 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))
8887anassrs 683 . . . . . . 7 ((((𝜑𝐹:𝑋1-1𝑌) ∧ 𝑧𝑈) ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))
8988ralrimivva 3073 . . . . . 6 (((𝜑𝐹:𝑋1-1𝑌) ∧ 𝑧𝑈) → ∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))
9089ex 449 . . . . 5 ((𝜑𝐹:𝑋1-1𝑌) → (𝑧𝑈 → ∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = )))
9168, 90sylbird 250 . . . 4 ((𝜑𝐹:𝑋1-1𝑌) → (𝑧 ∈ (Base‘𝐶) → ∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = )))
9291ralrimiv 3067 . . 3 ((𝜑𝐹:𝑋1-1𝑌) → ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))
931, 2, 3, 4, 8, 11, 13ismon2 16516 . . . 4 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))))
9493adantr 472 . . 3 ((𝜑𝐹:𝑋1-1𝑌) → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))))
9566, 92, 94mpbir2and 995 . 2 ((𝜑𝐹:𝑋1-1𝑌) → 𝐹 ∈ (𝑋𝑀𝑌))
9663, 95impbida 913 1 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋1-1𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  wral 3014  {csn 4285  cop 4291   × cxp 5216  ccom 5222   Fn wfn 5996  wf 5997  1-1wf1 5998  cfv 6001  (class class class)co 6765  Basecbs 15980  Hom chom 16075  compcco 16076  Catccat 16447  Monocmon 16510  SetCatcsetc 16847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-map 7976  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-3 11193  df-4 11194  df-5 11195  df-6 11196  df-7 11197  df-8 11198  df-9 11199  df-n0 11406  df-z 11491  df-dec 11607  df-uz 11801  df-fz 12441  df-struct 15982  df-ndx 15983  df-slot 15984  df-base 15986  df-hom 16089  df-cco 16090  df-cat 16451  df-cid 16452  df-mon 16512  df-setc 16848
This theorem is referenced by: (None)
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