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Theorem setcepi 18068

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 → 2_o ∈ 𝑈)

**Assertion**
Ref	Expression
setcepi	⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋–onto→𝑌))

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 2727	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2727	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2727	. . . . . 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 18065	. . . . . . 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 18058	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
11	9, 10	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		setcmon.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
13	12, 10	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	epihom 17716	. . . . 5 ⊢ (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
15	14	sselda 3978	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
16	6, 5, 2, 9, 12	elsetchom 18061	. . . . 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 590	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋⟶𝑌)
19	18	frnd 6724	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 ⊆ 𝑌)
20	18	ffnd 6717	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 Fn 𝑋)
21		fnfvelrn 7084	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ran 𝐹)
22	20, 21	sylan 579	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ran 𝐹)
23	22	iftrued 4532	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → if((𝐹‘𝑥) ∈ ran 𝐹, 1_o, ∅) = 1_o)
24	23	mpteq2dva 5242	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1_o, ∅)) = (𝑥 ∈ 𝑋 ↦ 1_o))
25	18	ffvelcdmda 7088	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝑌)
26	18	feqmptd 6961	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝑥)))
27		eqidd 2728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)))
28		eleq1 2816	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐹‘𝑥) → (𝑎 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
29	28	ifbid 4547	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐹‘𝑥) → if(𝑎 ∈ ran 𝐹, 1_o, ∅) = if((𝐹‘𝑥) ∈ ran 𝐹, 1_o, ∅))
30	25, 26, 27, 29	fmptco 7132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1_o, ∅)))
31		fconstmpt 5734	. . . . . . . . . . . . 13 ⊢ (𝑌 × {1_o}) = (𝑎 ∈ 𝑌 ↦ 1_o)
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1_o}) = (𝑎 ∈ 𝑌 ↦ 1_o))
33		eqidd 2728	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐹‘𝑥) → 1_o = 1_o)
34	25, 26, 32, 33	fmptco 7132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1_o}) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ 1_o))
35	24, 30, 34	3eqtr4d 2777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) ∘ 𝐹) = ((𝑌 × {1_o}) ∘ 𝐹))
36	5	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑈 ∈ 𝑉)
37	9	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ 𝑈)
38	12	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ 𝑈)
39		setcepi.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 2_o ∈ 𝑈)
40	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 2_o ∈ 𝑈)
41		eqid 2727	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅))
42		1oex 8490	. . . . . . . . . . . . . . . . 17 ⊢ 1_o ∈ V
43	42	prid2 4763	. . . . . . . . . . . . . . . 16 ⊢ 1_o ∈ {∅, 1_o}
44		df2o3 8488	. . . . . . . . . . . . . . . 16 ⊢ 2_o = {∅, 1_o}
45	43, 44	eleqtrri 2827	. . . . . . . . . . . . . . 15 ⊢ 1_o ∈ 2_o
46		0ex 5301	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
47	46	prid1 4762	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ {∅, 1_o}
48	47, 44	eleqtrri 2827	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ 2_o
49	45, 48	ifcli 4571	. . . . . . . . . . . . . 14 ⊢ if(𝑎 ∈ ran 𝐹, 1_o, ∅) ∈ 2_o
50	49	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑌 → if(𝑎 ∈ ran 𝐹, 1_o, ∅) ∈ 2_o)
51	41, 50	fmpti 7116	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)):𝑌⟶2_o
52	51	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)):𝑌⟶2_o)
53	6, 36, 3, 37, 38, 40, 18, 52	setcco 18063	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2_o)𝐹) = ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) ∘ 𝐹))
54		fconst6g 6780	. . . . . . . . . . . 12 ⊢ (1_o ∈ 2_o → (𝑌 × {1_o}):𝑌⟶2_o)
55	45, 54	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1_o}):𝑌⟶2_o)
56	6, 36, 3, 37, 38, 40, 18, 55	setcco 18063	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1_o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2_o)𝐹) = ((𝑌 × {1_o}) ∘ 𝐹))
57	35, 53, 56	3eqtr4d 2777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2_o)𝐹) = ((𝑌 × {1_o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2_o)𝐹))
58	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐶 ∈ Cat)
59	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ (Base‘𝐶))
60	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ (Base‘𝐶))
61	39, 10	eleqtrd 2830	. . . . . . . . . . 11 ⊢ (𝜑 → 2_o ∈ (Base‘𝐶))
62	61	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 2_o ∈ (Base‘𝐶))
63		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋𝐸𝑌))
64	6, 36, 2, 38, 40	elsetchom 18061	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) ∈ (𝑌(Hom ‘𝐶)2_o) ↔ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)):𝑌⟶2_o))
65	52, 64	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) ∈ (𝑌(Hom ‘𝐶)2_o))
66	6, 36, 2, 38, 40	elsetchom 18061	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1_o}) ∈ (𝑌(Hom ‘𝐶)2_o) ↔ (𝑌 × {1_o}):𝑌⟶2_o))
67	55, 66	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1_o}) ∈ (𝑌(Hom ‘𝐶)2_o))
68	1, 2, 3, 4, 58, 59, 60, 62, 63, 65, 67	epii 17717	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅))(⟨𝑋, 𝑌⟩(comp‘𝐶)2_o)𝐹) = ((𝑌 × {1_o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2_o)𝐹) ↔ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) = (𝑌 × {1_o})))
69	57, 68	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) = (𝑌 × {1_o}))
70	69, 31	eqtrdi 2783	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) = (𝑎 ∈ 𝑌 ↦ 1_o))
71	49	rgenw 3060	. . . . . . . 8 ⊢ ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1_o, ∅) ∈ 2_o
72		mpteqb 7018	. . . . . . . 8 ⊢ (∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1_o, ∅) ∈ 2_o → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) = (𝑎 ∈ 𝑌 ↦ 1_o) ↔ ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1_o, ∅) = 1_o))
73	71, 72	ax-mp 5	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1_o, ∅)) = (𝑎 ∈ 𝑌 ↦ 1_o) ↔ ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1_o, ∅) = 1_o)
74	70, 73	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1_o, ∅) = 1_o)
75		1n0 8502	. . . . . . . . . 10 ⊢ 1_o ≠ ∅
76	75	nesymi 2993	. . . . . . . . 9 ⊢ ¬ ∅ = 1_o
77		iffalse 4533	. . . . . . . . . 10 ⊢ (¬ 𝑎 ∈ ran 𝐹 → if(𝑎 ∈ ran 𝐹, 1_o, ∅) = ∅)
78	77	eqeq1d 2729	. . . . . . . . 9 ⊢ (¬ 𝑎 ∈ ran 𝐹 → (if(𝑎 ∈ ran 𝐹, 1_o, ∅) = 1_o ↔ ∅ = 1_o))
79	76, 78	mtbiri 327	. . . . . . . 8 ⊢ (¬ 𝑎 ∈ ran 𝐹 → ¬ if(𝑎 ∈ ran 𝐹, 1_o, ∅) = 1_o)
80	79	con4i 114	. . . . . . 7 ⊢ (if(𝑎 ∈ ran 𝐹, 1_o, ∅) = 1_o → 𝑎 ∈ ran 𝐹)
81	80	ralimi 3078	. . . . . 6 ⊢ (∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1_o, ∅) = 1_o → ∀𝑎 ∈ 𝑌 𝑎 ∈ ran 𝐹)
82	74, 81	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎 ∈ 𝑌 𝑎 ∈ ran 𝐹)
83		dfss3 3966	. . . . 5 ⊢ (𝑌 ⊆ ran 𝐹 ↔ ∀𝑎 ∈ 𝑌 𝑎 ∈ ran 𝐹)
84	82, 83	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ⊆ ran 𝐹)
85	19, 84	eqssd 3995	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 = 𝑌)
86		dffo2 6809	. . 3 ⊢ (𝐹:𝑋–onto→𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ran 𝐹 = 𝑌))
87	18, 85, 86	sylanbrc 582	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋–onto→𝑌)
88		fof 6805	. . . . 5 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
89	88	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝐹:𝑋⟶𝑌)
90	16	biimpar 477	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
91	89, 90	syldan 590	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
92	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝑈 = (Base‘𝐶))
93	92	eleq2d 2814	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → (𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (Base‘𝐶)))
94	5	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑈 ∈ 𝑉)
95	9	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑋 ∈ 𝑈)
96	12	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑌 ∈ 𝑈)
97		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑧 ∈ 𝑈)
98	89	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋⟶𝑌)
99		simprrl 780	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧))
100	6, 94, 2, 96, 97	elsetchom 18061	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ 𝑔:𝑌⟶𝑧))
101	99, 100	mpbid 231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔:𝑌⟶𝑧)
102	6, 94, 3, 95, 96, 97, 98, 101	setcco 18063	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (𝑔 ∘ 𝐹))
103		simprrr 781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ℎ ∈ (𝑌(Hom ‘𝐶)𝑧))
104	6, 94, 2, 96, 97	elsetchom 18061	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (ℎ ∈ (𝑌(Hom ‘𝐶)𝑧) ↔ ℎ:𝑌⟶𝑧))
105	103, 104	mpbid 231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ℎ:𝑌⟶𝑧)
106	6, 94, 3, 95, 96, 97, 98, 105	setcco 18063	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑧)𝐹) = (ℎ ∘ 𝐹))
107	102, 106	eqeq12d 2743	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (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)))
108		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋–onto→𝑌)
109	101	ffnd 6717	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 Fn 𝑌)
110	105	ffnd 6717	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ℎ Fn 𝑌)
111		cocan2 7295	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑔 Fn 𝑌 ∧ ℎ Fn 𝑌) → ((𝑔 ∘ 𝐹) = (ℎ ∘ 𝐹) ↔ 𝑔 = ℎ))
112	108, 109, 110, 111	syl3anc 1369	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔 ∘ 𝐹) = (ℎ ∘ 𝐹) ↔ 𝑔 = ℎ))
113	112	biimpd 228	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ((𝑔 ∘ 𝐹) = (ℎ ∘ 𝐹) → 𝑔 = ℎ))
114	107, 113	sylbid 239	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (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	anassrs 467	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ 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	ralrimivva 3195	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ 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	116	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → (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)))
118	93, 117	sylbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→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)))
119	118	ralrimiv 3140	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→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	1, 2, 3, 4, 8, 11, 13	isepi2 17715	. . . 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))))
121	120	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → (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))))
122	91, 119, 121	mpbir2and 712	. 2 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝐹 ∈ (𝑋𝐸𝑌))
123	87, 122	impbida 800	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋–onto→𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ⊆ wss 3944 ∅c0 4318 ifcif 4524 {csn 4624 {cpr 4626 ⟨cop 4630 ↦ cmpt 5225 × cxp 5670 ran crn 5673 ∘ ccom 5676 Fn wfn 6537 ⟶wf 6538 –onto→wfo 6540 ‘cfv 6542 (class class class)co 7414 1_oc1o 8473 2_oc2o 8474 Basecbs 17171 Hom chom 17235 compcco 17236 Catccat 17635 Epicepi 17703 SetCatcsetc 18055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-hom 17248 df-cco 17249 df-cat 17639 df-cid 17640 df-oppc 17683 df-mon 17704 df-epi 17705 df-setc 18056

This theorem is referenced by: (None)

W3C validator