Theorem setcmon 18263

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	⊢ (𝜑 → 𝑈 ∈ 𝑉)
setcmon.x	⊢ (𝜑 → 𝑋 ∈ 𝑈)
setcmon.y	⊢ (𝜑 → 𝑌 ∈ 𝑈)
setcmon.h	⊢ 𝑀 = (Mono‘𝐶)

Assertion

Ref	Expression
setcmon	⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋–1-1→𝑌))

Proof of Theorem setcmon

Dummy variables 𝑥 𝑔 ℎ 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2760	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		setcmon.h	. . . . . 6 ⊢ 𝑀 = (Mono‘𝐶)
5		setcmon.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
6		setcmon.c	. . . . . . . 8 ⊢ 𝐶 = (SetCat‘𝑈)
7	6	setccat 18261	. . . . . . 7 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
8	5, 7	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		setcmon.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
10	6, 5	setcbas 18254	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
11	9, 10	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		setcmon.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
13	12, 10	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	monhom 17911	. . . . 5 ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
15	14	sselda 3908	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
16	6, 5, 2, 9, 12	elsetchom 18257	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋⟶𝑌))
17	16	biimpa 482	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋⟶𝑌)
18	15, 17	syldan 603	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋⟶𝑌)
19		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
20	19	sneqd 4595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → {(𝐹‘𝑥)} = {(𝐹‘𝑦)})
21	20	xpeq2d 5680	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑋 × {(𝐹‘𝑥)}) = (𝑋 × {(𝐹‘𝑦)}))
22	18	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹:𝑋⟶𝑌)
23	22	ffnd 6710	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹 Fn 𝑋)
24		simprll 791	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 ∈ 𝑋)
25		fcoconst 7135	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹‘𝑥)}))
26	23, 24, 25	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹‘𝑥)}))
27		simprlr 792	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑦 ∈ 𝑋)
28		fcoconst 7135	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹‘𝑦)}))
29	23, 27, 28	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹‘𝑦)}))
30	21, 26, 29	3eqtr4d 2805	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑦})))
31	5	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑈 ∈ 𝑉)
32	9	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑋 ∈ 𝑈)
33	12	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑌 ∈ 𝑈)
34		fconst6g 6771	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 → (𝑋 × {𝑥}):𝑋⟶𝑋)
35	24, 34	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑋 × {𝑥}):𝑋⟶𝑋)
36	6, 31, 3, 32, 32, 33, 35, 22	setcco 18259	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑥})))
37		fconst6g 6771	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑋 → (𝑋 × {𝑦}):𝑋⟶𝑋)
38	27, 37	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑋 × {𝑦}):𝑋⟶𝑋)
39	6, 31, 3, 32, 32, 33, 38, 22	setcco 18259	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})) = (𝐹 ∘ (𝑋 × {𝑦})))
40	30, 36, 39	3eqtr4d 2805	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})))
41	8	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐶 ∈ Cat)
42	11	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑋 ∈ (Base‘𝐶))
43	13	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑌 ∈ (Base‘𝐶))
44		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹 ∈ (𝑋𝑀𝑌))
45	6, 31, 2, 32, 32	elsetchom 18257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑋 × {𝑥}) ∈ (𝑋(Hom ‘𝐶)𝑋) ↔ (𝑋 × {𝑥}):𝑋⟶𝑋))
46	35, 45	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑋 × {𝑥}) ∈ (𝑋(Hom ‘𝐶)𝑋))
47	6, 31, 2, 32, 32	elsetchom 18257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑋 × {𝑦}) ∈ (𝑋(Hom ‘𝐶)𝑋) ↔ (𝑋 × {𝑦}):𝑋⟶𝑋))
48	38, 47	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑋 × {𝑦}) ∈ (𝑋(Hom ‘𝐶)𝑋))
49	1, 2, 3, 4, 41, 42, 43, 42, 44, 46, 48	moni 17912	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑥})) = (𝐹(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝑋 × {𝑦})) ↔ (𝑋 × {𝑥}) = (𝑋 × {𝑦})))
50	40, 49	mpbid 235	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑋 × {𝑥}) = (𝑋 × {𝑦}))
51	50	fveq1d 6887	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑋 × {𝑥})‘𝑥) = ((𝑋 × {𝑦})‘𝑥))
52		vex 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
53	52	fvconst2 7210	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
54	24, 53	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
55		vex 3454	. . . . . . . 8 ⊢ 𝑦 ∈ V
56	55	fvconst2 7210	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
57	24, 56	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
58	51, 54, 57	3eqtr3d 2803	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 = 𝑦)
59	58	expr 462	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
60	59	ralrimivva 3205	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
61		dff13 7258	. . 3 ⊢ (𝐹:𝑋–1-1→𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
62	18, 60, 61	sylanbrc 595	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋–1-1→𝑌)
63		f1f 6778	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹:𝑋⟶𝑌)
64	16	biimpar 483	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
65	63, 64	sylan2 605	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
66	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → 𝑈 = (Base‘𝐶))
67	66	eleq2d 2846	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → (𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (Base‘𝐶)))
68	5	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑈 ∈ 𝑉)
69		simprl 783	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑧 ∈ 𝑈)
70	9	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑋 ∈ 𝑈)
71	12	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑌 ∈ 𝑈)
72		simprrl 793	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋))
73	6, 68, 2, 69, 70	elsetchom 18257	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↔ 𝑔:𝑧⟶𝑋))
74	72, 73	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑔:𝑧⟶𝑋)
75	63	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝐹:𝑋⟶𝑌)
76	6, 68, 3, 69, 70, 71, 74, 75	setcco 18259	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹 ∘ 𝑔))
77		simprrr 794	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))
78	6, 68, 2, 69, 70	elsetchom 18257	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (ℎ ∈ (𝑧(Hom ‘𝐶)𝑋) ↔ ℎ:𝑧⟶𝑋))
79	77, 78	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ℎ:𝑧⟶𝑋)
80	6, 68, 3, 69, 70, 71, 79, 75	setcco 18259	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) = (𝐹 ∘ ℎ))
81	76, 80	eqeq12d 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) ↔ (𝐹 ∘ 𝑔) = (𝐹 ∘ ℎ)))
82		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝐹:𝑋–1-1→𝑌)
83		cocan1 7299	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑔:𝑧⟶𝑋 ∧ ℎ:𝑧⟶𝑋) → ((𝐹 ∘ 𝑔) = (𝐹 ∘ ℎ) ↔ 𝑔 = ℎ))
84	82, 74, 79, 83	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹 ∘ 𝑔) = (𝐹 ∘ ℎ) ↔ 𝑔 = ℎ))
85	84	biimpd 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹 ∘ 𝑔) = (𝐹 ∘ ℎ) → 𝑔 = ℎ))
86	81, 85	sylbid 243	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → ((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))
87	86	anassrs 473	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑧 ∈ 𝑈) ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋))) → ((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))
88	87	ralrimivva 3205	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑧 ∈ 𝑈) → ∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))
89	88	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → (𝑧 ∈ 𝑈 → ∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ)))
90	67, 89	sylbird 263	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → (𝑧 ∈ (Base‘𝐶) → ∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ)))
91	90	ralrimiv 3153	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))
92	1, 2, 3, 4, 8, 11, 13	ismon2 17910	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))))
93	92	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)((𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))))
94	65, 91, 93	mpbir2and 726	. 2 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → 𝐹 ∈ (𝑋𝑀𝑌))
95	62, 94	impbida 813	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋–1-1→𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {csn 4583  ⟨cop 4589   × cxp 5648   ∘ ccom 5654   Fn wfn 6533  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7420  Basecbs 17388  Hom chom 17440  compcco 17441  Catccat 17839  Monocmon 17904  SetCatcsetc 18251

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11257  ax-resscn 11258  ax-1cn 11259  ax-icn 11260  ax-addcl 11261  ax-addrcl 11262  ax-mulcl 11263  ax-mulrcl 11264  ax-mulcom 11265  ax-addass 11266  ax-mulass 11267  ax-distr 11268  ax-i2m1 11269  ax-1ne0 11270  ax-1rid 11271  ax-rnegex 11272  ax-rrecex 11273  ax-cnre 11274  ax-pre-lttri 11275  ax-pre-lttrn 11276  ax-pre-ltadd 11277  ax-pre-mulgt0 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-er 8718  df-map 8850  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-pnf 11346  df-mnf 11347  df-xr 11348  df-ltxr 11349  df-le 11350  df-sub 11544  df-neg 11545  df-nn 12337  df-2 12406  df-3 12407  df-4 12408  df-5 12409  df-6 12410  df-7 12411  df-8 12412  df-9 12413  df-n0 12608  df-z 12695  df-dec 12816  df-uz 12967  df-fz 13641  df-struct 17326  df-slot 17361  df-ndx 17373  df-base 17389  df-hom 17453  df-cco 17454  df-cat 17843  df-cid 17844  df-mon 17906  df-setc 18252

This theorem is used by: (None)