Theorem setcepi 17200

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 2775	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2775	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2775	. . . . . 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 17197	. . . . . . 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 17190	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
11	9, 10	eleqtrd 2865	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12		setcmon.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
13	12, 10	eleqtrd 2865	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
14	1, 2, 3, 4, 8, 11, 13	epihom 16864	. . . . 5 ⊢ (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
15	14	sselda 3857	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
16	6, 5, 2, 9, 12	elsetchom 17193	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋⟶𝑌))
17	16	biimpa 469	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) → 𝐹:𝑋⟶𝑌)
18	15, 17	syldan 582	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋⟶𝑌)
19	18	frnd 6349	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 ⊆ 𝑌)
20	18	ffnd 6343	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 Fn 𝑋)
21		fnfvelrn 6671	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ran 𝐹)
22	20, 21	sylan 572	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ran 𝐹)
23	22	iftrued 4356	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅) = 1o)
24	23	mpteq2dva 5020	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅)) = (𝑥 ∈ 𝑋 ↦ 1o))
25	18	ffvelrnda 6674	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝑌)
26	18	feqmptd 6560	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝑥)))
27		eqidd 2776	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)))
28		eleq1 2850	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐹‘𝑥) → (𝑎 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
29	28	ifbid 4370	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐹‘𝑥) → if(𝑎 ∈ ran 𝐹, 1o, ∅) = if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅))
30	25, 26, 27, 29	fmptco 6712	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅)))
31		fconstmpt 5461	. . . . . . . . . . . . 13 ⊢ (𝑌 × {1o}) = (𝑎 ∈ 𝑌 ↦ 1o)
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) = (𝑎 ∈ 𝑌 ↦ 1o))
33		eqidd 2776	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝐹‘𝑥) → 1o = 1o)
34	25, 26, 32, 33	fmptco 6712	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ 1o))
35	24, 30, 34	3eqtr4d 2821	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
36	5	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑈 ∈ 𝑉)
37	9	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ 𝑈)
38	12	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ 𝑈)
39		setcepi.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 2o ∈ 𝑈)
40	39	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ 𝑈)
41		eqid 2775	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))
42		1oex 7909	. . . . . . . . . . . . . . . . 17 ⊢ 1o ∈ V
43	42	prid2 4571	. . . . . . . . . . . . . . . 16 ⊢ 1o ∈ {∅, 1o}
44		df2o3 7915	. . . . . . . . . . . . . . . 16 ⊢ 2o = {∅, 1o}
45	43, 44	eleqtrri 2862	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ 2o
46		0ex 5066	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
47	46	prid1 4570	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ {∅, 1o}
48	47, 44	eleqtrri 2862	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ 2o
49	45, 48	ifcli 4394	. . . . . . . . . . . . . 14 ⊢ if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
50	49	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑌 → if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o)
51	41, 50	fmpti 6697	. . . . . . . . . . . 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 17195	. . . . . . . . . 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 6395	. . . . . . . . . . . 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 17195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o})(⟨𝑋, 𝑌⟩(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o}) ∘ 𝐹))
57	35, 53, 56	3eqtr4d 2821	. . . . . . . . 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 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐶 ∈ Cat)
59	11	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑋 ∈ (Base‘𝐶))
60	13	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ∈ (Base‘𝐶))
61	39, 10	eleqtrd 2865	. . . . . . . . . . 11 ⊢ (𝜑 → 2o ∈ (Base‘𝐶))
62	61	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 2o ∈ (Base‘𝐶))
63		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋𝐸𝑌))
64	6, 36, 2, 38, 40	elsetchom 17193	. . . . . . . . . . 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 249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∈ (𝑌(Hom ‘𝐶)2o))
66	6, 36, 2, 38, 40	elsetchom 17193	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o) ↔ (𝑌 × {1o}):𝑌⟶2o))
67	55, 66	mpbird 249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) ∈ (𝑌(Hom ‘𝐶)2o))
68	1, 2, 3, 4, 58, 59, 60, 62, 63, 65, 67	epii 16865	. . . . . . . . 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 224	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑌 × {1o}))
70	69, 31	syl6eq 2827	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ 1o))
71	49	rgenw 3097	. . . . . . . 8 ⊢ ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈ 2o
72		mpteqb 6611	. . . . . . . 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 210	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ∀𝑎 ∈ 𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o)
75		1n0 7917	. . . . . . . . . 10 ⊢ 1o ≠ ∅
76	75	nesymi 3021	. . . . . . . . 9 ⊢ ¬ ∅ = 1o
77		iffalse 4357	. . . . . . . . . 10 ⊢ (¬ 𝑎 ∈ ran 𝐹 → if(𝑎 ∈ ran 𝐹, 1o, ∅) = ∅)
78	77	eqeq1d 2777	. . . . . . . . 9 ⊢ (¬ 𝑎 ∈ ran 𝐹 → (if(𝑎 ∈ ran 𝐹, 1o, ∅) = 1o ↔ ∅ = 1o))
79	76, 78	mtbiri 319	. . . . . . . 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 3107	. . . . . 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 3846	. . . . 5 ⊢ (𝑌 ⊆ ran 𝐹 ↔ ∀𝑎 ∈ 𝑌 𝑎 ∈ ran 𝐹)
84	82, 83	sylibr 226	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝑌 ⊆ ran 𝐹)
85	19, 84	eqssd 3874	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 = 𝑌)
86		dffo2 6421	. . 3 ⊢ (𝐹:𝑋–onto→𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ran 𝐹 = 𝑌))
87	18, 85, 86	sylanbrc 575	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹:𝑋–onto→𝑌)
88		fof 6417	. . . . 5 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
89	88	adantl 474	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝐹:𝑋⟶𝑌)
90	16	biimpar 470	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
91	89, 90	syldan 582	. . 3 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
92	10	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝑈 = (Base‘𝐶))
93	92	eleq2d 2848	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → (𝑧 ∈ 𝑈 ↔ 𝑧 ∈ (Base‘𝐶)))
94	5	ad2antrr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑈 ∈ 𝑉)
95	9	ad2antrr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑋 ∈ 𝑈)
96	12	ad2antrr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑌 ∈ 𝑈)
97		simprl 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑧 ∈ 𝑈)
98	89	adantr 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝐹:𝑋⟶𝑌)
99		simprrl 768	. . . . . . . . . . . 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 17193	. . . . . . . . . . . 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 224	. . . . . . . . . . 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 17195	. . . . . . . . . 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 769	. . . . . . . . . . . 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 17193	. . . . . . . . . . . 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 224	. . . . . . . . . . 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 17195	. . . . . . . . . 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 2790	. . . . . . . . 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 756	. . . . . . . . . . 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 6343	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 Fn 𝑌)
110	105	ffnd 6343	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ℎ Fn 𝑌)
111		cocan2 6871	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑔 Fn 𝑌 ∧ ℎ Fn 𝑌) → ((𝑔 ∘ 𝐹) = (ℎ ∘ 𝐹) ↔ 𝑔 = ℎ))
112	108, 109, 110, 111	syl3anc 1351	. . . . . . . . . 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 221	. . . . . . . . 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 232	. . . . . . . 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 460	. . . . . . 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 3138	. . . . . 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 405	. . . . 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 252	. . . 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 3128	. . 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 16863	. . . 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 473	. . 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 700	. 2 ⊢ ((𝜑 ∧ 𝐹:𝑋–onto→𝑌) → 𝐹 ∈ (𝑋𝐸𝑌))
123	87, 122	impbida 788	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋–onto→𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2048  ∀wral 3085   ⊆ wss 3828  ∅c0 4177  ifcif 4348  {csn 4439  {cpr 4441  ⟨cop 4445   ↦ cmpt 5006   × cxp 5402  ran crn 5405   ∘ ccom 5408   Fn wfn 6181  ⟶wf 6182  –onto→wfo 6184  ‘cfv 6186  (class class class)co 6974  1_oc1o 7894  2_oc2o 7895  Basecbs 16333  Hom chom 16426  compcco 16427  Catccat 16787  Epicepi 16851  SetCatcsetc 17187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-rep 5047  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277  ax-cnex 10387  ax-resscn 10388  ax-1cn 10389  ax-icn 10390  ax-addcl 10391  ax-addrcl 10392  ax-mulcl 10393  ax-mulrcl 10394  ax-mulcom 10395  ax-addass 10396  ax-mulass 10397  ax-distr 10398  ax-i2m1 10399  ax-1ne0 10400  ax-1rid 10401  ax-rnegex 10402  ax-rrecex 10403  ax-cnre 10404  ax-pre-lttri 10405  ax-pre-lttrn 10406  ax-pre-ltadd 10407  ax-pre-mulgt0 10408

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ne 2965  df-nel 3071  df-ral 3090  df-rex 3091  df-reu 3092  df-rmo 3093  df-rab 3094  df-v 3414  df-sbc 3681  df-csb 3786  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-pss 3844  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-tp 4444  df-op 4446  df-uni 4711  df-int 4748  df-iun 4792  df-br 4928  df-opab 4990  df-mpt 5007  df-tr 5029  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7498  df-2nd 7499  df-tpos 7692  df-wrecs 7747  df-recs 7809  df-rdg 7847  df-1o 7901  df-2o 7902  df-oadd 7905  df-er 8085  df-map 8204  df-en 8303  df-dom 8304  df-sdom 8305  df-fin 8306  df-pnf 10472  df-mnf 10473  df-xr 10474  df-ltxr 10475  df-le 10476  df-sub 10668  df-neg 10669  df-nn 11436  df-2 11500  df-3 11501  df-4 11502  df-5 11503  df-6 11504  df-7 11505  df-8 11506  df-9 11507  df-n0 11705  df-z 11791  df-dec 11909  df-uz 12056  df-fz 12706  df-struct 16335  df-ndx 16336  df-slot 16337  df-base 16339  df-sets 16340  df-hom 16439  df-cco 16440  df-cat 16791  df-cid 16792  df-oppc 16834  df-mon 16852  df-epi 16853  df-setc 17188

This theorem is referenced by: (None)