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Theorem setcmon 18083

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 2728	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2728	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2728	. . . . . 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 18081	. . . . . . 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 18074	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
11	9, 10	eleqtrd 2831	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		setcmon.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
13	12, 10	eleqtrd 2831	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	monhom 17725	. . . . 5 ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
15	14	sselda 3982	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
16	6, 5, 2, 9, 12	elsetchom 18077	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋⟶𝑌))
17	16	biimpa 475	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋⟶𝑌)
18	15, 17	syldan 589	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋⟶𝑌)
19		simprr 771	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
20	19	sneqd 4644	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → {(𝐹‘𝑥)} = {(𝐹‘𝑦)})
21	20	xpeq2d 5712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑋 × {(𝐹‘𝑥)}) = (𝑋 × {(𝐹‘𝑦)}))
22	18	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹:𝑋⟶𝑌)
23	22	ffnd 6728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹 Fn 𝑋)
24		simprll 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 ∈ 𝑋)
25		fcoconst 7149	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹‘𝑥)}))
26	23, 24, 25	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝑋 × {(𝐹‘𝑥)}))
27		simprlr 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑦 ∈ 𝑋)
28		fcoconst 7149	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹‘𝑦)}))
29	23, 27, 28	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹 ∘ (𝑋 × {𝑦})) = (𝑋 × {(𝐹‘𝑦)}))
30	21, 26, 29	3eqtr4d 2778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹 ∘ (𝑋 × {𝑥})) = (𝐹 ∘ (𝑋 × {𝑦})))
31	5	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑈 ∈ 𝑉)
32	9	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑋 ∈ 𝑈)
33	12	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑌 ∈ 𝑈)
34		fconst6g 6791	. . . . . . . . . . 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 18079	. . . . . . . . 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 6791	. . . . . . . . . . 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 18079	. . . . . . . . 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 2778	. . . . . . . 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 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐶 ∈ Cat)
42	11	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑋 ∈ (Base‘𝐶))
43	13	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑌 ∈ (Base‘𝐶))
44		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹 ∈ (𝑋𝑀𝑌))
45	6, 31, 2, 32, 32	elsetchom 18077	. . . . . . . . . 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 256	. . . . . . . . 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 18077	. . . . . . . . . 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 256	. . . . . . . . 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 17726	. . . . . . . 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 231	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑋 × {𝑥}) = (𝑋 × {𝑦}))
51	50	fveq1d 6904	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑋 × {𝑥})‘𝑥) = ((𝑋 × {𝑦})‘𝑥))
52		vex 3477	. . . . . . . 8 ⊢ 𝑥 ∈ V
53	52	fvconst2 7222	. . . . . . 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 3477	. . . . . . . 8 ⊢ 𝑦 ∈ V
56	55	fvconst2 7222	. . . . . . 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 2776	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 = 𝑦)
59	58	expr 455	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
60	59	ralrimivva 3198	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
61		dff13 7271	. . 3 ⊢ (𝐹:𝑋–1-1→𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
62	18, 60, 61	sylanbrc 581	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝑀𝑌)) → 𝐹:𝑋–1-1→𝑌)
63		f1f 6798	. . . 4 ⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹:𝑋⟶𝑌)
64	16	biimpar 476	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
65	63, 64	sylan2 591	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
66	10	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → 𝑈 = (Base‘𝐶))
67	66	eleq2d 2815	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → (𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (Base‘𝐶)))
68	5	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑈 ∈ 𝑉)
69		simprl 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑧 ∈ 𝑈)
70	9	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑋 ∈ 𝑈)
71	12	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑋)))) → 𝑌 ∈ 𝑈)
72		simprrl 779	. . . . . . . . . . . 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 18077	. . . . . . . . . . . 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 231	. . . . . . . . . . 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 725	. . . . . . . . . . 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 18079	. . . . . . . . . 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 780	. . . . . . . . . . . 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 18077	. . . . . . . . . . . 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 231	. . . . . . . . . . 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 18079	. . . . . . . . . 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 2744	. . . . . . . . 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 767	. . . . . . . . . . 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 7306	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑔:𝑧⟶𝑋 ∧ ℎ:𝑧⟶𝑋) → ((𝐹 ∘ 𝑔) = (𝐹 ∘ ℎ) ↔ 𝑔 = ℎ))
84	82, 74, 79, 83	syl3anc 1368	. . . . . . . . . 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 228	. . . . . . . . 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 239	. . . . . . . 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 466	. . . . . . 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 3198	. . . . . 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 411	. . . . 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 259	. . . 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 3142	. . 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 17724	. . . 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 479	. . 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 711	. 2 ⊢ ((𝜑 ∧ 𝐹:𝑋–1-1→𝑌) → 𝐹 ∈ (𝑋𝑀𝑌))
95	62, 94	impbida 799	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋–1-1→𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 {csn 4632 ⟨cop 4638 × cxp 5680 ∘ ccom 5686 Fn wfn 6548 ⟶wf 6549 –1-1→wf1 6550 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 Hom chom 17251 compcco 17252 Catccat 17651 Monocmon 17718 SetCatcsetc 18071

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 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-hom 17264 df-cco 17265 df-cat 17655 df-cid 17656 df-mon 17720 df-setc 18072

This theorem is referenced by: (None)

W3C validator