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Theorem setcmon 17342
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 2826 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2826 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2826 . . . . . 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 17340 . . . . . . 7 (𝑈𝑉𝐶 ∈ Cat)
85, 7syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
9 setcmon.x . . . . . . 7 (𝜑𝑋𝑈)
106, 5setcbas 17333 . . . . . . 7 (𝜑𝑈 = (Base‘𝐶))
119, 10eleqtrd 2920 . . . . . 6 (𝜑𝑋 ∈ (Base‘𝐶))
12 setcmon.y . . . . . . 7 (𝜑𝑌𝑈)
1312, 10eleqtrd 2920 . . . . . 6 (𝜑𝑌 ∈ (Base‘𝐶))
141, 2, 3, 4, 8, 11, 13monhom 17000 . . . . 5 (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1514sselda 3971 . . . 4 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
166, 5, 2, 9, 12elsetchom 17336 . . . . 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 769 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
2019sneqd 4576 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → {(𝐹𝑥)} = {(𝐹𝑦)})
2120xpeq2d 5584 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {(𝐹𝑥)}) = (𝑋 × {(𝐹𝑦)}))
2218adantr 481 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹:𝑋𝑌)
2322ffnd 6514 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹 Fn 𝑋)
24 simprll 775 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥𝑋)
25 fcoconst 6894 . . . . . . . . . . 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 776 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑦𝑋)
28 fcoconst 6894 . . . . . . . . . . 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 2871 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑦})))
315ad2antrr 722 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑈𝑉)
329ad2antrr 722 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑋𝑈)
3312ad2antrr 722 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑌𝑈)
34 fconst6g 6567 . . . . . . . . . . 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 17338 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑥})))
37 fconst6g 6567 . . . . . . . . . . 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 17338 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})) = (𝐹 ∘ (𝑋 × {𝑦})))
4030, 36, 393eqtr4d 2871 . . . . . . . 8 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})))
418ad2antrr 722 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐶 ∈ Cat)
4211ad2antrr 722 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑋 ∈ (Base‘𝐶))
4313ad2antrr 722 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑌 ∈ (Base‘𝐶))
44 simplr 765 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹 ∈ (𝑋𝑀𝑌))
456, 31, 2, 32, 32elsetchom 17336 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥}) ∈ (𝑋(Hom ‘𝐶)𝑋) ↔ (𝑋 × {𝑥}):𝑋𝑋))
4635, 45mpbird 258 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑥}) ∈ (𝑋(Hom ‘𝐶)𝑋))
476, 31, 2, 32, 32elsetchom 17336 . . . . . . . . . 10 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑦}) ∈ (𝑋(Hom ‘𝐶)𝑋) ↔ (𝑋 × {𝑦}):𝑋𝑋))
4838, 47mpbird 258 . . . . . . . . 9 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑦}) ∈ (𝑋(Hom ‘𝐶)𝑋))
491, 2, 3, 4, 41, 42, 43, 42, 44, 46, 48moni 17001 . . . . . . . 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 233 . . . . . . 7 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑋 × {𝑥}) = (𝑋 × {𝑦}))
5150fveq1d 6671 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥})‘𝑥) = ((𝑋 × {𝑦})‘𝑥))
52 vex 3503 . . . . . . . 8 𝑥 ∈ V
5352fvconst2 6964 . . . . . . 7 (𝑥𝑋 → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
5424, 53syl 17 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
55 vex 3503 . . . . . . . 8 𝑦 ∈ V
5655fvconst2 6964 . . . . . . 7 (𝑥𝑋 → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
5724, 56syl 17 . . . . . 6 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
5851, 54, 573eqtr3d 2869 . . . . 5 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥𝑋𝑦𝑋) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥 = 𝑦)
5958expr 457 . . . 4 (((𝜑𝐹 ∈ (𝑋𝑀𝑌)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6059ralrimivva 3196 . . 3 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
61 dff13 7009 . . 3 (𝐹:𝑋1-1𝑌 ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6218, 60, 61sylanbrc 583 . 2 ((𝜑𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋1-1𝑌)
63 f1f 6574 . . . 4 (𝐹:𝑋1-1𝑌𝐹:𝑋𝑌)
6416biimpar 478 . . . 4 ((𝜑𝐹:𝑋𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
6563, 64sylan2 592 . . 3 ((𝜑𝐹:𝑋1-1𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
6610adantr 481 . . . . . 6 ((𝜑𝐹:𝑋1-1𝑌) → 𝑈 = (Base‘𝐶))
6766eleq2d 2903 . . . . 5 ((𝜑𝐹:𝑋1-1𝑌) → (𝑧𝑈𝑧 ∈ (Base‘𝐶)))
685ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑈𝑉)
69 simprl 767 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑧𝑈)
709ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑋𝑈)
7112ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑌𝑈)
72 simprrl 777 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋))
736, 68, 2, 69, 70elsetchom 17336 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↔ 𝑔:𝑧𝑋))
7472, 73mpbid 233 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑔:𝑧𝑋)
7563ad2antlr 723 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝐹:𝑋𝑌)
766, 68, 3, 69, 70, 71, 74, 75setcco 17338 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹𝑔))
77 simprrr 778 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ∈ (𝑧(Hom ‘𝐶)𝑋))
786, 68, 2, 69, 70elsetchom 17336 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ( ∈ (𝑧(Hom ‘𝐶)𝑋) ↔ :𝑧𝑋))
7977, 78mpbid 233 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → :𝑧𝑋)
806, 68, 3, 69, 70, 71, 79, 75setcco 17338 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)) = (𝐹))
8176, 80eqeq12d 2842 . . . . . . . . 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 765 . . . . . . . . . . 11 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝐹:𝑋1-1𝑌)
83 cocan1 7043 . . . . . . . . . . 11 ((𝐹:𝑋1-1𝑌𝑔:𝑧𝑋:𝑧𝑋) → ((𝐹𝑔) = (𝐹) ↔ 𝑔 = ))
8482, 74, 79, 83syl3anc 1365 . . . . . . . . . 10 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹𝑔) = (𝐹) ↔ 𝑔 = ))
8584biimpd 230 . . . . . . . . 9 (((𝜑𝐹:𝑋1-1𝑌) ∧ (𝑧𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹𝑔) = (𝐹) → 𝑔 = ))
8681, 85sylbid 241 . . . . . . . 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 3196 . . . . . 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 261 . . . 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 3186 . . 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 16999 . . . 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 709 . 2 ((𝜑𝐹:𝑋1-1𝑌) → 𝐹 ∈ (𝑋𝑀𝑌))
9562, 94impbida 797 1 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋1-1𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  {csn 4564  cop 4570   × cxp 5552  ccom 5558   Fn wfn 6349  wf 6350  1-1wf1 6351  cfv 6354  (class class class)co 7150  Basecbs 16478  Hom chom 16571  compcco 16572  Catccat 16930  Monocmon 16993  SetCatcsetc 17330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12888  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-hom 16584  df-cco 16585  df-cat 16934  df-cid 16935  df-mon 16995  df-setc 17331
This theorem is referenced by: (None)
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