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Theorem setcmon 17002
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 2765 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2765 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2765 . . . . . 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 17000 . . . . . . 7 (𝑈𝑉𝐶 ∈ Cat)
85, 7syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
9 setcmon.x . . . . . . 7 (𝜑𝑋𝑈)
106, 5setcbas 16993 . . . . . . 7 (𝜑𝑈 = (Base‘𝐶))
119, 10eleqtrd 2846 . . . . . 6 (𝜑𝑋 ∈ (Base‘𝐶))
12 setcmon.y . . . . . . 7 (𝜑𝑌𝑈)
1312, 10eleqtrd 2846 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐶))
141, 2, 3, 4, 8, 11, 13monhom 16660 . . . . 5 (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1514sselda 3761 . . . 4 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
166, 5, 2, 9, 12elsetchom 16996 . . . . 5 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋𝑌))
1716biimpa 468 . . . 4 ((𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋𝑌)
1815, 17syldan 585 . . 3 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋𝑌)
19 simprr 789 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
2019sneqd 4346 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → {(𝐹𝑥)} = {(𝐹𝑦)})
2120xpeq2d 5307 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {(𝐹𝑥)}) = (𝑋 × {(𝐹𝑦)}))
2218adantr 472 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹:𝑋𝑌)
2322ffnd 6224 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹 Fn 𝑋)
24 simprll 797 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥𝑋)
25 fcoconst 6592 . . . . . . . . . . 11 ((𝐹 Fn 𝑋𝑥𝑋) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹𝑥)}))
2623, 24, 25syl2anc 579 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹𝑥)}))
27 simprlr 798 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑦𝑋)
28 fcoconst 6592 . . . . . . . . . . 11 ((𝐹 Fn 𝑋𝑦𝑋) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹𝑦)}))
2923, 27, 28syl2anc 579 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹𝑦)}))
3021, 26, 293eqtr4d 2809 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑦})))
315ad2antrr 717 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑈𝑉)
329ad2antrr 717 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑋𝑈)
3312ad2antrr 717 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑌𝑈)
34 fconst6g 6276 . . . . . . . . . . 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 16998 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑥})))
37 fconst6g 6276 . . . . . . . . . . 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 16998 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})) = (𝐹 ∘ (𝑋 × {𝑦})))
4030, 36, 393eqtr4d 2809 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})))
418ad2antrr 717 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐶 ∈ Cat)
4211ad2antrr 717 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑋 ∈ (Base‘𝐶))
4313ad2antrr 717 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑌 ∈ (Base‘𝐶))
44 simplr 785 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹 ∈ (𝑋𝑀𝑌))
456, 31, 2, 32, 32elsetchom 16996 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥}) ∈ (𝑋(Hom ‘𝐶)𝑋) ↔ (𝑋 × {𝑥}):𝑋𝑋))
4635, 45mpbird 248 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑥}) ∈ (𝑋(Hom ‘𝐶)𝑋))
476, 31, 2, 32, 32elsetchom 16996 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑦}) ∈ (𝑋(Hom ‘𝐶)𝑋) ↔ (𝑋 × {𝑦}):𝑋𝑋))
4838, 47mpbird 248 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑦}) ∈ (𝑋(Hom ‘𝐶)𝑋))
491, 2, 3, 4, 41, 42, 43, 42, 44, 46, 48moni 16661 . . . . . . . 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 223 . . . . . . 7 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑥}) = (𝑋 × {𝑦}))
5150fveq1d 6377 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥})‘𝑥) = ((𝑋 × {𝑦})‘𝑥))
52 vex 3353 . . . . . . . 8 𝑥 ∈ V
5352fvconst2 6662 . . . . . . 7 (𝑥𝑋 → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
5424, 53syl 17 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
55 vex 3353 . . . . . . . 8 𝑦 ∈ V
5655fvconst2 6662 . . . . . . 7 (𝑥𝑋 → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
5724, 56syl 17 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
5851, 54, 573eqtr3d 2807 . . . . 5 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥 = 𝑦)
5958expr 448 . . . 4 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6059ralrimivva 3118 . . 3 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
61 dff13 6704 . . 3 (𝐹:𝑋1-1𝑌 ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6218, 60, 61sylanbrc 578 . 2 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋1-1𝑌)
63 f1f 6283 . . . 4 (𝐹:𝑋1-1𝑌𝐹:𝑋𝑌)
6416biimpar 469 . . . 4 ((𝜑𝐹:𝑋𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
6563, 64sylan2 586 . . 3 ((𝜑𝐹:𝑋1-1𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
6610adantr 472 . . . . . 6 ((𝜑𝐹:𝑋1-1𝑌) → 𝑈 = (Base‘𝐶))
6766eleq2d 2830 . . . . 5 ((𝜑𝐹:𝑋1-1𝑌) → (𝑧𝑈𝑧 ∈ (Base‘𝐶)))
685ad2antrr 717 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑈𝑉)
69 simprl 787 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑧𝑈)
709ad2antrr 717 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑋𝑈)
7112ad2antrr 717 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑌𝑈)
72 simprrl 799 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋))
736, 68, 2, 69, 70elsetchom 16996 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↔ 𝑔:𝑧𝑋))
7472, 73mpbid 223 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑔:𝑧𝑋)
7563ad2antlr 718 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝐹:𝑋𝑌)
766, 68, 3, 69, 70, 71, 74, 75setcco 16998 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹𝑔))
77 simprrr 800 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ∈ (𝑧(Hom ‘𝐶)𝑋))
786, 68, 2, 69, 70elsetchom 16996 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ( ∈ (𝑧(Hom ‘𝐶)𝑋) ↔ :𝑧𝑋))
7977, 78mpbid 223 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → :𝑧𝑋)
806, 68, 3, 69, 70, 71, 79, 75setcco 16998 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) = (𝐹))
8176, 80eqeq12d 2780 . . . . . . . . 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 785 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝐹:𝑋1-1𝑌)
83 cocan1 6738 . . . . . . . . . . 11 ((𝐹:𝑋1-1𝑌𝑔:𝑧𝑋:𝑧𝑋) → ((𝐹𝑔) = (𝐹) ↔ 𝑔 = ))
8482, 74, 79, 83syl3anc 1490 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹𝑔) = (𝐹) ↔ 𝑔 = ))
8584biimpd 220 . . . . . . . . 9 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹𝑔) = (𝐹) → 𝑔 = ))
8681, 85sylbid 231 . . . . . . . 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 459 . . . . . . 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 3118 . . . . . 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 401 . . . . 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 251 . . . 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 3112 . . 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 16659 . . . 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 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 = ))))
9465, 91, 93mpbir2and 704 . 2 ((𝜑𝐹:𝑋1-1𝑌) → 𝐹 ∈ (𝑋𝑀𝑌))
9562, 94impbida 835 1 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋1-1𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  {csn 4334  cop 4340   × cxp 5275  ccom 5281   Fn wfn 6063  wf 6064  1-1wf1 6065  cfv 6068  (class class class)co 6842  Basecbs 16130  Hom chom 16225  compcco 16226  Catccat 16590  Monocmon 16653  SetCatcsetc 16990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-hom 16238  df-cco 16239  df-cat 16594  df-cid 16595  df-mon 16655  df-setc 16991
This theorem is referenced by: (None)
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