Theorem setcepi 17801

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–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 2740	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2740	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2740	. . . . . 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 17798	. . . . . . 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 17791	. . . . . . 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	epihom 17452	. . . . 5 ⊢ (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
15	14	sselda 3926	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
16	6, 5, 2, 9, 12	elsetchom 17794	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋⟶𝑌))
17	16	biimpa 477	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋⟶𝑌)
18	15, 17	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋⟶𝑌)
19	18	frnd 6606	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 ⊆ 𝑌)
20	18	ffnd 6599	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 Fn 𝑋)
21		fnfvelrn 6955	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ran 𝐹)
22	20, 21	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ran 𝐹)
23	22	iftrued 4473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅) = 1o)
24	23	mpteq2dva 5179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅)) = (𝑥 ∈ 𝑋 ↦ 1o))
25	18	ffvelrnda 6958	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝑌)
26	18	feqmptd 6834	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝑥)))
27		eqidd 2741	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)))
28		eleq1 2828	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐹‘𝑥) → (𝑎 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
29	28	ifbid 4488	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐹‘𝑥) → if(𝑎 ∈ ran 𝐹, 1o, ∅) = if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅))
30	25, 26, 27, 29	fmptco 6998	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅)))
31		fconstmpt 5650	. . . . . . . . . . . . 13 ⊢ (𝑌 × {1o}) = (𝑎 ∈ 𝑌 ↦ 1o)
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) = (𝑎 ∈ 𝑌 ↦ 1o))
33		eqidd 2741	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐹‘𝑥) → 1o = 1o)
34	25, 26, 32, 33	fmptco 6998	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ 1o))
35	24, 30, 34	3eqtr4d 2790	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
36	5	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑈 ∈ 𝑉)
37	9	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ 𝑈)
38	12	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ 𝑈)
39		setcepi.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 2o ∈ 𝑈)
40	39	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ 𝑈)
41		eqid 2740	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))
42		1oex 8298	. . . . . . . . . . . . . . . . 17 ⊢ 1o ∈ V
43	42	prid2 4705	. . . . . . . . . . . . . . . 16 ⊢ 1o ∈ {∅, 1o}
44		df2o3 8296	. . . . . . . . . . . . . . . 16 ⊢ 2o = {∅, 1o}
45	43, 44	eleqtrri 2840	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ 2o
46		0ex 5235	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
47	46	prid1 4704	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ {∅, 1o}
48	47, 44	eleqtrri 2840	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ 2o
49	45, 48	ifcli 4512	. . . . . . . . . . . . . 14 ⊢ if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
50	49	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑌 → if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o)
51	41, 50	fmpti 6983	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o
52	51	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o)
53	6, 36, 3, 37, 38, 40, 18, 52	setcco 17796	. . . . . . . . . 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))
54		fconst6g 6661	. . . . . . . . . . . 12 ⊢ (1o ∈ 2o → (𝑌 × {1o}):𝑌⟶2o)
55	45, 54	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}):𝑌⟶2o)
56	6, 36, 3, 37, 38, 40, 18, 55	setcco 17796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
57	35, 53, 56	3eqtr4d 2790	. . . . . . . . 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))
58	8	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐶 ∈ Cat)
59	11	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ (Base‘𝐶))
60	13	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ (Base‘𝐶))
61	39, 10	eleqtrd 2843	. . . . . . . . . . 11 ⊢ (𝜑 → 2o ∈ (Base‘𝐶))
62	61	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ (Base‘𝐶))
63		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋𝐸𝑌))
64	6, 36, 2, 38, 40	elsetchom 17794	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)):𝑌⟶2o))
65	52, 64	mpbird 256	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o))
66	6, 36, 2, 38, 40	elsetchom 17794	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑌 × {1o}):𝑌⟶2o))
67	55, 66	mpbird 256	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o))
68	1, 2, 3, 4, 58, 59, 60, 62, 63, 65, 67	epii 17453	. . . . . . . . 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})))
69	57, 68	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑌 × {1o}))
70	69, 31	eqtrdi 2796	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ 1o))
71	49	rgenw 3078	. . . . . . . 8 ⊢ ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
72		mpteqb 6891	. . . . . . . 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))
73	71, 72	ax-mp 5	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ 1o) ↔ ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
74	70, 73	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
75		1n0 8309	. . . . . . . . . 10 ⊢ 1o ≠ ∅
76	75	nesymi 3003	. . . . . . . . 9 ⊢ ¬ ∅ = 1o
77		iffalse 4474	. . . . . . . . . 10 ⊢ (¬ 𝑎 ∈ ran 𝐹 → if(𝑎 ∈ ran 𝐹, 1o, ∅) = ∅)
78	77	eqeq1d 2742	. . . . . . . . 9 ⊢ (¬ 𝑎 ∈ ran 𝐹 → (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o ↔ ∅ = 1o))
79	76, 78	mtbiri 327	. . . . . . . 8 ⊢ (¬ 𝑎 ∈ ran 𝐹 → ¬ if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
80	79	con4i 114	. . . . . . 7 ⊢ (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o → 𝑎 ∈ ran 𝐹)
81	80	ralimi 3089	. . . . . 6 ⊢ (∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o → ∀𝑎 ∈ 𝑌 𝑎 ∈ ran 𝐹)
82	74, 81	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎 ∈ 𝑌 𝑎 ∈ ran 𝐹)
83		dfss3 3914	. . . . 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 3943	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 = 𝑌)
86		dffo2 6690	. . 3 ⊢ (𝐹:𝑋–onto→𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ran 𝐹 = 𝑌))
87	18, 85, 86	sylanbrc 583	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋–onto→𝑌)
88		fof 6686	. . . . 5 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
89	88	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝐹:𝑋⟶𝑌)
90	16	biimpar 478	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
91	89, 90	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
92	10	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝑈 = (Base‘𝐶))
93	92	eleq2d 2826	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → (𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (Base‘𝐶)))
94	5	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑈 ∈ 𝑉)
95	9	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑋 ∈ 𝑈)
96	12	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑌 ∈ 𝑈)
97		simprl 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑧 ∈ 𝑈)
98	89	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋⟶𝑌)
99		simprrl 778	. . . . . . . . . . . 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 17794	. . . . . . . . . . . 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 17796	. . . . . . . . . 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 779	. . . . . . . . . . . 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 17794	. . . . . . . . . . . 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 17796	. . . . . . . . . 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 2756	. . . . . . . . 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 766	. . . . . . . . . . 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 6599	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 Fn 𝑌)
110	105	ffnd 6599	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ℎ Fn 𝑌)
111		cocan2 7160	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑔 Fn 𝑌 ∧ ℎ Fn 𝑌) → ((𝑔 ∘ 𝐹) = (ℎ ∘ 𝐹) ↔ 𝑔 = ℎ))
112	108, 109, 110, 111	syl3anc 1370	. . . . . . . . . 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 468	. . . . . . 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 3117	. . . . . 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 413	. . . . 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 259	. . . 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 3109	. . 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 17451	. . . 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 481	. . 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 710	. 2 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝐹 ∈ (𝑋𝐸𝑌))
123	87, 122	impbida 798	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋–onto→𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ∀wral 3066   ⊆ wss 3892  ∅c0 4262  ifcif 4465  {csn 4567  {cpr 4569  ⟨cop 4573   ↦ cmpt 5162   × cxp 5588  ran crn 5591   ∘ ccom 5594   Fn wfn 6427  ⟶wf 6428  –onto→wfo 6430  ‘cfv 6432  (class class class)co 7271  1_oc1o 8281  2_oc2o 8282  Basecbs 16910  Hom chom 16971  compcco 16972  Catccat 17371  Epicepi 17439  SetCatcsetc 17788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-tpos 8033  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-2o 8289  df-er 8481  df-map 8600  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12437  df-uz 12582  df-fz 13239  df-struct 16846  df-sets 16863  df-slot 16881  df-ndx 16893  df-base 16911  df-hom 16984  df-cco 16985  df-cat 17375  df-cid 17376  df-oppc 17419  df-mon 17440  df-epi 17441  df-setc 17789

This theorem is referenced by: (None)