Theorem setcmon 18043

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 18041	. . . . . . 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 18034	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
11	9, 10	eleqtrd 2839	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		setcmon.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
13	12, 10	eleqtrd 2839	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	monhom 17691	. . . . 5 ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
15	14	sselda 3922	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
16	6, 5, 2, 9, 12	elsetchom 18037	. . . . 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 592	. . 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 4580	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → {(𝐹‘𝑥)} = {(𝐹‘𝑦)})
21	20	xpeq2d 5652	. . . . . . . . . 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 6661	. . . . . . . . . . 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 7079	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹‘𝑥)}))
26	23, 24, 25	syl2anc 585	. . . . . . . . . 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 7079	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹‘𝑦)}))
29	23, 27, 28	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹‘𝑦)}))
30	21, 26, 29	3eqtr4d 2782	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑦})))
31	5	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑈 ∈ 𝑉)
32	9	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑋 ∈ 𝑈)
33	12	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑌 ∈ 𝑈)
34		fconst6g 6721	. . . . . . . . . . 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 18039	. . . . . . . . 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 6721	. . . . . . . . . . 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 18039	. . . . . . . . 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 2782	. . . . . . . 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 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐶 ∈ Cat)
42	11	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑋 ∈ (Base‘𝐶))
43	13	ad2antrr 727	. . . . . . . . 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 18037	. . . . . . . . . 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 18037	. . . . . . . . . 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 17692	. . . . . . . 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 6834	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑋 × {𝑥})‘𝑥) = ((𝑋 × {𝑦})‘𝑥))
52		vex 3434	. . . . . . . 8 ⊢ 𝑥 ∈ V
53	52	fvconst2 7150	. . . . . . 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 3434	. . . . . . . 8 ⊢ 𝑦 ∈ V
56	55	fvconst2 7150	. . . . . . 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 2780	. . . . 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 3181	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
61		dff13 7200	. . 3 ⊢ (𝐹:𝑋–1-1→𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
62	18, 60, 61	sylanbrc 584	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋–1-1→𝑌)
63		f1f 6728	. . . 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 594	. . 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 2823	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → (𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (Base‘𝐶)))
68	5	ad2antrr 727	. . . . . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑋 ∈ 𝑈)
71	12	ad2antrr 727	. . . . . . . . . . 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 18037	. . . . . . . . . . . 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 728	. . . . . . . . . . 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 18039	. . . . . . . . . 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 18037	. . . . . . . . . . . 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 18039	. . . . . . . . . 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 7237	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑔:𝑧⟶𝑋 ∧ ℎ:𝑧⟶𝑋) → ((𝐹 ∘ 𝑔) = (𝐹 ∘ ℎ) ↔ 𝑔 = ℎ))
84	82, 74, 79, 83	syl3anc 1374	. . . . . . . . . 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 3181	. . . . . 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 3129	. . 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 17690	. . . 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 714	. 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 1542   ∈ wcel 2114  ∀wral 3052  {csn 4568  ⟨cop 4574   × cxp 5620   ∘ ccom 5626   Fn wfn 6485  ⟶wf 6486  –1-1→wf1 6487  ‘cfv 6490  (class class class)co 7358  Basecbs 17168  Hom chom 17220  compcco 17221  Catccat 17619  Monocmon 17684  SetCatcsetc 18031

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-3 12234  df-4 12235  df-5 12236  df-6 12237  df-7 12238  df-8 12239  df-9 12240  df-n0 12427  df-z 12514  df-dec 12634  df-uz 12778  df-fz 13451  df-struct 17106  df-slot 17141  df-ndx 17153  df-base 17169  df-hom 17233  df-cco 17234  df-cat 17623  df-cid 17624  df-mon 17686  df-setc 18032

This theorem is referenced by: (None)