Theorem setcepi 17987

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 2730	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2730	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2730	. . . . . 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 17984	. . . . . . 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 17977	. . . . . . 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	epihom 17641	. . . . 5 ⊢ (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
15	14	sselda 3932	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
16	6, 5, 2, 9, 12	elsetchom 17980	. . . . 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 591	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋⟶𝑌)
19	18	frnd 6655	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 ⊆ 𝑌)
20	18	ffnd 6648	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 Fn 𝑋)
21		fnfvelrn 7008	. . . . . . . . . . . . . 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 4481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅) = 1o)
24	23	mpteq2dva 5182	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅)) = (𝑥 ∈ 𝑋 ↦ 1o))
25	18	ffvelcdmda 7012	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝑌)
26	18	feqmptd 6885	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝑥)))
27		eqidd 2731	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)))
28		eleq1 2817	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐹‘𝑥) → (𝑎 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
29	28	ifbid 4497	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐹‘𝑥) → if(𝑎 ∈ ran 𝐹, 1o, ∅) = if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅))
30	25, 26, 27, 29	fmptco 7057	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅)))
31		fconstmpt 5676	. . . . . . . . . . . . 13 ⊢ (𝑌 × {1o}) = (𝑎 ∈ 𝑌 ↦ 1o)
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) = (𝑎 ∈ 𝑌 ↦ 1o))
33		eqidd 2731	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐹‘𝑥) → 1o = 1o)
34	25, 26, 32, 33	fmptco 7057	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ 1o))
35	24, 30, 34	3eqtr4d 2775	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = ((𝑌 × {1o}) ∘ 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 → 2o ∈ 𝑈)
40	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ 𝑈)
41		eqid 2730	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))
42		1oex 8390	. . . . . . . . . . . . . . . . 17 ⊢ 1o ∈ V
43	42	prid2 4714	. . . . . . . . . . . . . . . 16 ⊢ 1o ∈ {∅, 1o}
44		df2o3 8388	. . . . . . . . . . . . . . . 16 ⊢ 2o = {∅, 1o}
45	43, 44	eleqtrri 2828	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ 2o
46		0ex 5243	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
47	46	prid1 4713	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ {∅, 1o}
48	47, 44	eleqtrri 2828	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ 2o
49	45, 48	ifcli 4521	. . . . . . . . . . . . . 14 ⊢ if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
50	49	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑌 → if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o)
51	41, 50	fmpti 7040	. . . . . . . . . . . 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 17982	. . . . . . . . . 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 6708	. . . . . . . . . . . 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 17982	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
57	35, 53, 56	3eqtr4d 2775	. . . . . . . . 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 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 2831	. . . . . . . . . . 11 ⊢ (𝜑 → 2o ∈ (Base‘𝐶))
62	61	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ (Base‘𝐶))
63		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋𝐸𝑌))
64	6, 36, 2, 38, 40	elsetchom 17980	. . . . . . . . . . 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 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o))
66	6, 36, 2, 38, 40	elsetchom 17980	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑌 × {1o}):𝑌⟶2o))
67	55, 66	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o))
68	1, 2, 3, 4, 58, 59, 60, 62, 63, 65, 67	epii 17642	. . . . . . . . 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 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑌 × {1o}))
70	69, 31	eqtrdi 2781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ 1o))
71	49	rgenw 3049	. . . . . . . 8 ⊢ ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
72		mpteqb 6943	. . . . . . . 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 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
75		1n0 8398	. . . . . . . . . 10 ⊢ 1o ≠ ∅
76	75	nesymi 2983	. . . . . . . . 9 ⊢ ¬ ∅ = 1o
77		iffalse 4482	. . . . . . . . . 10 ⊢ (¬ 𝑎 ∈ ran 𝐹 → if(𝑎 ∈ ran 𝐹, 1o, ∅) = ∅)
78	77	eqeq1d 2732	. . . . . . . . 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 3067	. . . . . 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 3921	. . . . 5 ⊢ (𝑌 ⊆ ran 𝐹 ↔ ∀𝑎 ∈ 𝑌 𝑎 ∈ ran 𝐹)
84	82, 83	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ⊆ ran 𝐹)
85	19, 84	eqssd 3950	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 = 𝑌)
86		dffo2 6735	. . 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 6731	. . . . 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 591	. . 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 2815	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → (𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (Base‘𝐶)))
94	5	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑈 ∈ 𝑉)
95	9	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑋 ∈ 𝑈)
96	12	ad2antrr 726	. . . . . . . . . . 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 17980	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 17982	. . . . . . . . . 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 17980	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 17982	. . . . . . . . . 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 2746	. . . . . . . . 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 6648	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 Fn 𝑌)
110	105	ffnd 6648	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ℎ Fn 𝑌)
111		cocan2 7221	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑔 Fn 𝑌 ∧ ℎ Fn 𝑌) → ((𝑔 ∘ 𝐹) = (ℎ ∘ 𝐹) ↔ 𝑔 = ℎ))
112	108, 109, 110, 111	syl3anc 1373	. . . . . . . . . 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 229	. . . . . . . . 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 240	. . . . . . . 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 3173	. . . . . 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 3121	. . 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 17640	. . . 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 713	. 2 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝐹 ∈ (𝑋𝐸𝑌))
123	87, 122	impbida 800	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋–onto→𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∀wral 3045   ⊆ wss 3900  ∅c0 4281  ifcif 4473  {csn 4574  {cpr 4576  ⟨cop 4580   ↦ cmpt 5170   × cxp 5612  ran crn 5615   ∘ ccom 5618   Fn wfn 6472  ⟶wf 6473  –onto→wfo 6475  ‘cfv 6477  (class class class)co 7341  1_oc1o 8373  2_oc2o 8374  Basecbs 17112  Hom chom 17164  compcco 17165  Catccat 17562  Epicepi 17628  SetCatcsetc 17974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-tpos 8151  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-7 12185  df-8 12186  df-9 12187  df-n0 12374  df-z 12461  df-dec 12581  df-uz 12725  df-fz 13400  df-struct 17050  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-hom 17177  df-cco 17178  df-cat 17566  df-cid 17567  df-oppc 17610  df-mon 17629  df-epi 17630  df-setc 17975

This theorem is referenced by: (None)