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Theorem setcmon 17973
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 2736 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2736 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2736 . . . . . 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 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, 13monhom 17618 . . . . 5 (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1514sselda 3944 . . . 4 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → 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 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋𝑌)
19 simprr 771 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
2019sneqd 4598 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → {(𝐹𝑥)} = {(𝐹𝑦)})
2120xpeq2d 5663 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {(𝐹𝑥)}) = (𝑋 × {(𝐹𝑦)}))
2218adantr 481 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹:𝑋𝑌)
2322ffnd 6669 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹 Fn 𝑋)
24 simprll 777 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥𝑋)
25 fcoconst 7080 . . . . . . . . . . 11 ((𝐹 Fn 𝑋𝑥𝑋) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹𝑥)}))
2623, 24, 25syl2anc 584 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹𝑥)}))
27 simprlr 778 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑦𝑋)
28 fcoconst 7080 . . . . . . . . . . 11 ((𝐹 Fn 𝑋𝑦𝑋) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹𝑦)}))
2923, 27, 28syl2anc 584 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹𝑦)}))
3021, 26, 293eqtr4d 2786 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑦})))
315ad2antrr 724 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑈𝑉)
329ad2antrr 724 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑋𝑈)
3312ad2antrr 724 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑌𝑈)
34 fconst6g 6731 . . . . . . . . . . 11 (𝑥𝑋 → (𝑋 × {𝑥}):𝑋𝑋)
3524, 34syl 17 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑥}):𝑋𝑋)
366, 31, 3, 32, 32, 33, 35, 22setcco 17969 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑥})))
37 fconst6g 6731 . . . . . . . . . . 11 (𝑦𝑋 → (𝑋 × {𝑦}):𝑋𝑋)
3827, 37syl 17 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑦}):𝑋𝑋)
396, 31, 3, 32, 32, 33, 38, 22setcco 17969 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})) = (𝐹 ∘ (𝑋 × {𝑦})))
4030, 36, 393eqtr4d 2786 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})))
418ad2antrr 724 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐶 ∈ Cat)
4211ad2antrr 724 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑋 ∈ (Base‘𝐶))
4313ad2antrr 724 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑌 ∈ (Base‘𝐶))
44 simplr 767 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹 ∈ (𝑋𝑀𝑌))
456, 31, 2, 32, 32elsetchom 17967 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥}) ∈ (𝑋(Hom ‘𝐶)𝑋) ↔ (𝑋 × {𝑥}):𝑋𝑋))
4635, 45mpbird 256 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑥}) ∈ (𝑋(Hom ‘𝐶)𝑋))
476, 31, 2, 32, 32elsetchom 17967 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑦}) ∈ (𝑋(Hom ‘𝐶)𝑋) ↔ (𝑋 × {𝑦}):𝑋𝑋))
4838, 47mpbird 256 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑦}) ∈ (𝑋(Hom ‘𝐶)𝑋))
491, 2, 3, 4, 41, 42, 43, 42, 44, 46, 48moni 17619 . . . . . . . 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})))
5040, 49mpbid 231 . . . . . . 7 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑥}) = (𝑋 × {𝑦}))
5150fveq1d 6844 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥})‘𝑥) = ((𝑋 × {𝑦})‘𝑥))
52 vex 3449 . . . . . . . 8 𝑥 ∈ V
5352fvconst2 7153 . . . . . . 7 (𝑥𝑋 → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
5424, 53syl 17 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
55 vex 3449 . . . . . . . 8 𝑦 ∈ V
5655fvconst2 7153 . . . . . . 7 (𝑥𝑋 → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
5724, 56syl 17 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
5851, 54, 573eqtr3d 2784 . . . . 5 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥 = 𝑦)
5958expr 457 . . . 4 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6059ralrimivva 3197 . . 3 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
61 dff13 7202 . . 3 (𝐹:𝑋1-1𝑌 ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6218, 60, 61sylanbrc 583 . 2 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋1-1𝑌)
63 f1f 6738 . . . 4 (𝐹:𝑋1-1𝑌𝐹:𝑋𝑌)
6416biimpar 478 . . . 4 ((𝜑𝐹:𝑋𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
6563, 64sylan2 593 . . 3 ((𝜑𝐹:𝑋1-1𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
6610adantr 481 . . . . . 6 ((𝜑𝐹:𝑋1-1𝑌) → 𝑈 = (Base‘𝐶))
6766eleq2d 2823 . . . . 5 ((𝜑𝐹:𝑋1-1𝑌) → (𝑧𝑈𝑧 ∈ (Base‘𝐶)))
685ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑈𝑉)
69 simprl 769 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑧𝑈)
709ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑋𝑈)
7112ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑌𝑈)
72 simprrl 779 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋))
736, 68, 2, 69, 70elsetchom 17967 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↔ 𝑔:𝑧𝑋))
7472, 73mpbid 231 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑔:𝑧𝑋)
7563ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝐹:𝑋𝑌)
766, 68, 3, 69, 70, 71, 74, 75setcco 17969 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹𝑔))
77 simprrr 780 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ∈ (𝑧(Hom ‘𝐶)𝑋))
786, 68, 2, 69, 70elsetchom 17967 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ( ∈ (𝑧(Hom ‘𝐶)𝑋) ↔ :𝑧𝑋))
7977, 78mpbid 231 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → :𝑧𝑋)
806, 68, 3, 69, 70, 71, 79, 75setcco 17969 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) = (𝐹))
8176, 80eqeq12d 2752 . . . . . . . . 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)))
82 simplr 767 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝐹:𝑋1-1𝑌)
83 cocan1 7237 . . . . . . . . . . 11 ((𝐹:𝑋1-1𝑌𝑔:𝑧𝑋:𝑧𝑋) → ((𝐹𝑔) = (𝐹) ↔ 𝑔 = ))
8482, 74, 79, 83syl3anc 1371 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹𝑔) = (𝐹) ↔ 𝑔 = ))
8584biimpd 228 . . . . . . . . 9 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹𝑔) = (𝐹) → 𝑔 = ))
8681, 85sylbid 239 . . . . . . . 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 = ))
8786anassrs 468 . . . . . . 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 = ))
8887ralrimivva 3197 . . . . . 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 = ))
8988ex 413 . . . . 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 = )))
9067, 89sylbird 259 . . . 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 = )))
9190ralrimiv 3142 . . 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 = ))
921, 2, 3, 4, 8, 11, 13ismon2 17617 . . . 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 = ))))
9392adantr 481 . . 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 = ))))
9465, 91, 93mpbir2and 711 . 2 ((𝜑𝐹:𝑋1-1𝑌) → 𝐹 ∈ (𝑋𝑀𝑌))
9562, 94impbida 799 1 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋1-1𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  {csn 4586  cop 4592   × cxp 5631  ccom 5637   Fn wfn 6491  wf 6492  1-1wf1 6493  cfv 6496  (class class class)co 7357  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Monocmon 17611  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-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  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-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-mon 17613  df-setc 17962
This theorem is referenced by: (None)
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