Theorem setcmon 18132

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 2737	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2737	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2737	. . . . . 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 18130	. . . . . . 7 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
8	5, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		setcmon.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
10	6, 5	setcbas 18123	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
11	9, 10	eleqtrd 2843	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		setcmon.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
13	12, 10	eleqtrd 2843	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	monhom 17779	. . . . 5 ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
15	14	sselda 3983	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
16	6, 5, 2, 9, 12	elsetchom 18126	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋⟶𝑌))
17	16	biimpa 476	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋⟶𝑌)
18	15, 17	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋⟶𝑌)
19		simprr 773	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
20	19	sneqd 4638	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → {(𝐹‘𝑥)} = {(𝐹‘𝑦)})
21	20	xpeq2d 5715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑋 × {(𝐹‘𝑥)}) = (𝑋 × {(𝐹‘𝑦)}))
22	18	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹:𝑋⟶𝑌)
23	22	ffnd 6737	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹 Fn 𝑋)
24		simprll 779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 ∈ 𝑋)
25		fcoconst 7154	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹‘𝑥)}))
26	23, 24, 25	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹‘𝑥)}))
27		simprlr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑦 ∈ 𝑋)
28		fcoconst 7154	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹‘𝑦)}))
29	23, 27, 28	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹‘𝑦)}))
30	21, 26, 29	3eqtr4d 2787	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑦})))
31	5	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑈 ∈ 𝑉)
32	9	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑋 ∈ 𝑈)
33	12	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑌 ∈ 𝑈)
34		fconst6g 6797	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 → (𝑋 × {𝑥}):𝑋⟶𝑋)
35	24, 34	syl 17	. . . . . . . . . 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 18128	. . . . . . . . 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 6797	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑋 → (𝑋 × {𝑦}):𝑋⟶𝑋)
38	27, 37	syl 17	. . . . . . . . . 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 18128	. . . . . . . . 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 2787	. . . . . . . 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 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐶 ∈ Cat)
42	11	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑋 ∈ (Base‘𝐶))
43	13	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑌 ∈ (Base‘𝐶))
44		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹 ∈ (𝑋𝑀𝑌))
45	6, 31, 2, 32, 32	elsetchom 18126	. . . . . . . . . 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 257	. . . . . . . . 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 18126	. . . . . . . . . 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 257	. . . . . . . . 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 17780	. . . . . . . 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 232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑋 × {𝑥}) = (𝑋 × {𝑦}))
51	50	fveq1d 6908	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑋 × {𝑥})‘𝑥) = ((𝑋 × {𝑦})‘𝑥))
52		vex 3484	. . . . . . . 8 ⊢ 𝑥 ∈ V
53	52	fvconst2 7224	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
54	24, 53	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑋 × {𝑥})‘𝑥) = 𝑥)
55		vex 3484	. . . . . . . 8 ⊢ 𝑦 ∈ V
56	55	fvconst2 7224	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
57	24, 56	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑋 × {𝑦})‘𝑥) = 𝑦)
58	51, 54, 57	3eqtr3d 2785	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 = 𝑦)
59	58	expr 456	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
60	59	ralrimivva 3202	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
61		dff13 7275	. . 3 ⊢ (𝐹:𝑋–1-1→𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
62	18, 60, 61	sylanbrc 583	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋–1-1→𝑌)
63		f1f 6804	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹:𝑋⟶𝑌)
64	16	biimpar 477	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
65	63, 64	sylan2 593	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
66	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → 𝑈 = (Base‘𝐶))
67	66	eleq2d 2827	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → (𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (Base‘𝐶)))
68	5	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑈 ∈ 𝑉)
69		simprl 771	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑧 ∈ 𝑈)
70	9	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑋 ∈ 𝑈)
71	12	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑌 ∈ 𝑈)
72		simprrl 781	. . . . . . . . . . . 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 18126	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 727	. . . . . . . . . . 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 18128	. . . . . . . . . 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 782	. . . . . . . . . . . 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 18126	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 18128	. . . . . . . . . 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 2753	. . . . . . . . 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 769	. . . . . . . . . . 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 7311	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑔:𝑧⟶𝑋 ∧ ℎ:𝑧⟶𝑋) → ((𝐹 ∘ 𝑔) = (𝐹 ∘ ℎ) ↔ 𝑔 = ℎ))
84	82, 74, 79, 83	syl3anc 1373	. . . . . . . . . 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 229	. . . . . . . . 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 240	. . . . . . . 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 467	. . . . . . 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 3202	. . . . . 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 412	. . . . 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 260	. . . 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 3145	. . 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 17778	. . . 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 480	. . 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 713	. 2 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → 𝐹 ∈ (𝑋𝑀𝑌))
95	62, 94	impbida 801	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋–1-1→𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {csn 4626  ⟨cop 4632   × cxp 5683   ∘ ccom 5689   Fn wfn 6556  ⟶wf 6557  –1-1→wf1 6558  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Monocmon 17772  SetCatcsetc 18120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-mon 17774  df-setc 18121

This theorem is referenced by: (None)