Theorem setcepi 18292

Description: An epimorphism of sets is a surjection. (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)
setcepi.h	⊢ 𝐸 = (Epi‘𝐶)
setcepi.2	⊢ (𝜑 → 2o ∈ 𝑈)

Assertion

Ref	Expression
setcepi	⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋⟶onto𝑌))

Proof of Theorem setcepi

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

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2763	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2763	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		setcepi.h	. . . . . 6 ⊢ 𝐸 = (Epi‘𝐶)
5		setcmon.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
6		setcmon.c	. . . . . . . 8 ⊢ 𝐶 = (SetCat‘𝑈)
7	6	setccat 18289	. . . . . . 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 18282	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
11	9, 10	eleqtrd 2866	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		setcmon.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
13	12, 10	eleqtrd 2866	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	epihom 17946	. . . . 5 ⊢ (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
15	14	sselda 3913	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
16	6, 5, 2, 9, 12	elsetchom 18285	. . . . 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 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋⟶𝑌)
19	18	frnd 6740	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 ⊆ 𝑌)
20	18	ffnd 6732	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 Fn 𝑋)
21		fnfvelrn 7104	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ran 𝐹)
22	20, 21	sylan 592	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ran 𝐹)
23	22	iftrued 4493	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅) = 1o)
24	23	mpteq2dva 5201	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅)) = (𝑥 ∈ 𝑋 ↦ 1o))
25	18	ffvelcdmda 7108	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝑌)
26	18	feqmptd 6976	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝑥)))
27		eqidd 2764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)))
28		eleq1 2852	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐹‘𝑥) → (𝑎 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
29	28	ifbid 4509	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐹‘𝑥) → if(𝑎 ∈ ran 𝐹, 1o, ∅) = if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅))
30	25, 26, 27, 29	fmptco 7154	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅)))
31		fconstmpt 5717	. . . . . . . . . . . . 13 ⊢ (𝑌 × {1o}) = (𝑎 ∈ 𝑌 ↦ 1o)
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) = (𝑎 ∈ 𝑌 ↦ 1o))
33		eqidd 2764	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐹‘𝑥) → 1o = 1o)
34	25, 26, 32, 33	fmptco 7154	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ 1o))
35	24, 30, 34	3eqtr4d 2808	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
36	5	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑈 ∈ 𝑉)
37	9	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ 𝑈)
38	12	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ 𝑈)
39		setcepi.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 2o ∈ 𝑈)
40	39	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ 𝑈)
41		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))
42		1oelpr 8515	. . . . . . . . . . . . . . . 16 ⊢ 1o ∈ {∅, 1o}
43		df2o3 8512	. . . . . . . . . . . . . . . 16 ⊢ 2o = {∅, 1o}
44	42, 43	eleqtrri 2863	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ 2o
45		0ex 5264	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
46	45	prid1 4723	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ {∅, 1o}
47	46, 43	eleqtrri 2863	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ 2o
48	44, 47	ifcli 4533	. . . . . . . . . . . . . 14 ⊢ if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
49	48	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑌 → if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o)
50	41, 49	fmpti 7136	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o
51	50	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o)
52	6, 36, 3, 37, 38, 40, 18, 51	setcco 18287	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹))
53		fconst6g 6793	. . . . . . . . . . . 12 ⊢ (1o ∈ 2o → (𝑌 × {1o}):𝑌⟶2o)
54	44, 53	mp1i 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}):𝑌⟶2o)
55	6, 36, 3, 37, 38, 40, 18, 54	setcco 18287	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
56	35, 52, 55	3eqtr4d 2808	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹))
57	8	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐶 ∈ Cat)
58	11	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ (Base‘𝐶))
59	13	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ (Base‘𝐶))
60	39, 10	eleqtrd 2866	. . . . . . . . . . 11 ⊢ (𝜑 → 2o ∈ (Base‘𝐶))
61	60	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ (Base‘𝐶))
62		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋𝐸𝑌))
63	6, 36, 2, 38, 40	elsetchom 18285	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o))
64	51, 63	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o))
65	6, 36, 2, 38, 40	elsetchom 18285	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑌 × {1o}):𝑌⟶2o))
66	54, 65	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o))
67	1, 2, 3, 4, 57, 58, 59, 61, 62, 64, 66	epii 17947	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) ↔ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑌 × {1o})))
68	56, 67	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑌 × {1o}))
69	68, 31	eqtrdi 2814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ 1o))
70	48	rgenw 3084	. . . . . . . 8 ⊢ ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
71		mpteqb 7036	. . . . . . . 8 ⊢ (∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ 1o) ↔ ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o))
72	70, 71	ax-mp 5	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ 1o) ↔ ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
73	69, 72	sylib 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
74		1n0 8523	. . . . . . . . . 10 ⊢ 1o ≠ ∅
75	74	nesymi 3016	. . . . . . . . 9 ⊢ ¬ ∅ = 1o
76		iffalse 4494	. . . . . . . . . 10 ⊢ (¬ 𝑎 ∈ ran 𝐹 → if(𝑎 ∈ ran 𝐹, 1o, ∅) = ∅)
77	76	eqeq1d 2765	. . . . . . . . 9 ⊢ (¬ 𝑎 ∈ ran 𝐹 → (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o ↔ ∅ = 1o))
78	75, 77	mtbiri 330	. . . . . . . 8 ⊢ (¬ 𝑎 ∈ ran 𝐹 → ¬ if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
79	78	con4i 115	. . . . . . 7 ⊢ (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o → 𝑎 ∈ ran 𝐹)
80	79	ralimi 3103	. . . . . 6 ⊢ (∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o → ∀𝑎 ∈ 𝑌 𝑎 ∈ ran 𝐹)
81	73, 80	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎 ∈ 𝑌 𝑎 ∈ ran 𝐹)
82		dfss3 3902	. . . . 5 ⊢ (𝑌 ⊆ ran 𝐹 ↔ ∀𝑎 ∈ 𝑌 𝑎 ∈ ran 𝐹)
83	81, 82	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ⊆ ran 𝐹)
84	19, 83	eqssd 3930	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 = 𝑌)
85		dffo2 6822	. . 3 ⊢ (𝐹:𝑋⟶onto𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ran 𝐹 = 𝑌))
86	18, 84, 85	sylanbrc 595	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋⟶onto𝑌)
87		fof 6818	. . . . 5 ⊢ (𝐹:𝑋⟶onto𝑌 → 𝐹:𝑋⟶𝑌)
88	87	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) → 𝐹:𝑋⟶𝑌)
89	16	biimpar 483	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
90	88, 89	syldan 603	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
91	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) → 𝑈 = (Base‘𝐶))
92	91	eleq2d 2850	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) → (𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (Base‘𝐶)))
93	5	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑈 ∈ 𝑉)
94	9	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑋 ∈ 𝑈)
95	12	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑌 ∈ 𝑈)
96		simprl 783	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑧 ∈ 𝑈)
97	88	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋⟶𝑌)
98		simprrl 793	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧))
99	6, 93, 2, 95, 96	elsetchom 18285	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ 𝑔:𝑌⟶𝑧))
100	98, 99	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔:𝑌⟶𝑧)
101	6, 93, 3, 94, 95, 96, 97, 100	setcco 18287	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝑔 ∘ 𝐹))
102		simprrr 794	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))
103	6, 93, 2, 95, 96	elsetchom 18285	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (ℎ ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ ℎ:𝑌⟶𝑧))
104	102, 103	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ℎ:𝑌⟶𝑧)
105	6, 93, 3, 94, 95, 96, 97, 104	setcco 18287	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (ℎ ∘ 𝐹))
106	101, 105	eqeq12d 2779	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) ↔ (𝑔 ∘ 𝐹) = (ℎ ∘ 𝐹)))
107		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋⟶onto𝑌)
108	100	ffnd 6732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 Fn 𝑌)
109	104	ffnd 6732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ℎ Fn 𝑌)
110		cocan2 7323	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋⟶onto𝑌 ∧ 𝑔 Fn 𝑌 ∧ ℎ Fn 𝑌) → ((𝑔 ∘ 𝐹) = (ℎ ∘ 𝐹) ↔ 𝑔 = ℎ))
111	107, 108, 109, 110	syl3anc 1399	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔 ∘ 𝐹) = (ℎ ∘ 𝐹) ↔ 𝑔 = ℎ))
112	111	biimpd 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔 ∘ 𝐹) = (ℎ ∘ 𝐹) → 𝑔 = ℎ))
113	106, 112	sylbid 243	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ℎ))
114	113	anassrs 473	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝑧 ∈ 𝑈) ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))) → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ℎ))
115	114	ralrimivva 3209	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) ∧ 𝑧 ∈ 𝑈) → ∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ℎ))
116	115	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) → (𝑧 ∈ 𝑈 → ∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ℎ)))
117	92, 116	sylbird 263	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) → (𝑧 ∈ (Base‘𝐶) → ∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ℎ)))
118	117	ralrimiv 3157	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) → ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ℎ))
119	1, 2, 3, 4, 8, 11, 13	isepi2 17945	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ℎ))))
120	119	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) → 𝑔 = ℎ))))
121	90, 118, 120	mpbir2and 726	. 2 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶onto𝑌) → 𝐹 ∈ (𝑋𝐸𝑌))
122	86, 121	impbida 813	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋⟶onto𝑌))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  ∅c0 4281  ifcif 4485  {csn 4588  {cpr 4590  ⟨cop 4838   ↦ cmpt 5189   × cxp 5653  ran crn 5656   ∘ ccom 5659   Fn wfn 6553  ⟶wf 6554  ⟶ontowfo 6556  ‘cfv 6558  (class class class)co 7443  1_oc1o 8497  2_oc2o 8498  Basecbs 17416  Hom chom 17468  compcco 17469  Catccat 17867  Epicepi 17933  SetCatcsetc 18279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-tpos 8271  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-hom 17481  df-cco 17482  df-cat 17871  df-cid 17872  df-oppc 17915  df-mon 17934  df-epi 17935  df-setc 18280

This theorem is used by: (None)