Step | Hyp | Ref
| Expression |
1 | | eqid 2738 |
. . . . . 6
⊢
(Base‘𝐶) =
(Base‘𝐶) |
2 | | eqid 2738 |
. . . . . 6
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
3 | | eqid 2738 |
. . . . . 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 17800 |
. . . . . . 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 17793 |
. . . . . . 7
⊢ (𝜑 → 𝑈 = (Base‘𝐶)) |
11 | 9, 10 | eleqtrd 2841 |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
12 | | setcmon.y |
. . . . . . 7
⊢ (𝜑 → 𝑌 ∈ 𝑈) |
13 | 12, 10 | eleqtrd 2841 |
. . . . . 6
⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
14 | 1, 2, 3, 4, 8, 11,
13 | epihom 17454 |
. . . . 5
⊢ (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌)) |
15 | 14 | sselda 3921 |
. . . 4
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
16 | 6, 5, 2, 9, 12 | elsetchom 17796 |
. . . . 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 6608 |
. . . 4
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 ⊆ 𝑌) |
20 | 18 | ffnd 6601 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 Fn 𝑋) |
21 | | fnfvelrn 6958 |
. . . . . . . . . . . . . 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 4467 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅) =
1o) |
24 | 23 | mpteq2dva 5174 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅)) = (𝑥 ∈ 𝑋 ↦ 1o)) |
25 | 18 | ffvelrnda 6961 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ 𝑌) |
26 | 18 | feqmptd 6837 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐹‘𝑥))) |
27 | | eqidd 2739 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅))) |
28 | | eleq1 2826 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝐹‘𝑥) → (𝑎 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹)) |
29 | 28 | ifbid 4482 |
. . . . . . . . . . . 12
⊢ (𝑎 = (𝐹‘𝑥) → if(𝑎 ∈ ran 𝐹, 1o, ∅) = if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅)) |
30 | 25, 26, 27, 29 | fmptco 7001 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ if((𝐹‘𝑥) ∈ ran 𝐹, 1o, ∅))) |
31 | | fconstmpt 5649 |
. . . . . . . . . . . . 13
⊢ (𝑌 × {1o}) =
(𝑎 ∈ 𝑌 ↦ 1o) |
32 | 31 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑌 × {1o}) = (𝑎 ∈ 𝑌 ↦ 1o)) |
33 | | eqidd 2739 |
. . . . . . . . . . . 12
⊢ (𝑎 = (𝐹‘𝑥) → 1o =
1o) |
34 | 25, 26, 32, 33 | fmptco 7001 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o}) ∘ 𝐹) = (𝑥 ∈ 𝑋 ↦ 1o)) |
35 | 24, 30, 34 | 3eqtr4d 2788 |
. . . . . . . . . 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 2738 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) |
42 | | 1oex 8307 |
. . . . . . . . . . . . . . . . 17
⊢
1o ∈ V |
43 | 42 | prid2 4699 |
. . . . . . . . . . . . . . . 16
⊢
1o ∈ {∅, 1o} |
44 | | df2o3 8305 |
. . . . . . . . . . . . . . . 16
⊢
2o = {∅, 1o} |
45 | 43, 44 | eleqtrri 2838 |
. . . . . . . . . . . . . . 15
⊢
1o ∈ 2o |
46 | | 0ex 5231 |
. . . . . . . . . . . . . . . . 17
⊢ ∅
∈ V |
47 | 46 | prid1 4698 |
. . . . . . . . . . . . . . . 16
⊢ ∅
∈ {∅, 1o} |
48 | 47, 44 | eleqtrri 2838 |
. . . . . . . . . . . . . . 15
⊢ ∅
∈ 2o |
49 | 45, 48 | ifcli 4506 |
. . . . . . . . . . . . . 14
⊢ if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈
2o |
50 | 49 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∈ 𝑌 → if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈
2o) |
51 | 41, 50 | fmpti 6986 |
. . . . . . . . . . . 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 17798 |
. . . . . . . . . 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 6663 |
. . . . . . . . . . . 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 17798 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ((𝑌 × {1o})(〈𝑋, 𝑌〉(comp‘𝐶)2o)𝐹) = ((𝑌 × {1o}) ∘ 𝐹)) |
57 | 35, 53, 56 | 3eqtr4d 2788 |
. . . . . . . . 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 2841 |
. . . . . . . . . . 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 17796 |
. . . . . . . . . . 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 17796 |
. . . . . . . . . . 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 17455 |
. . . . . . . . 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 2794 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → (𝑎 ∈ 𝑌 ↦ if(𝑎 ∈ ran 𝐹, 1o, ∅)) = (𝑎 ∈ 𝑌 ↦ 1o)) |
71 | 49 | rgenw 3076 |
. . . . . . . 8
⊢
∀𝑎 ∈
𝑌 if(𝑎 ∈ ran 𝐹, 1o, ∅) ∈
2o |
72 | | mpteqb 6894 |
. . . . . . . 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 8318 |
. . . . . . . . . 10
⊢
1o ≠ ∅ |
76 | 75 | nesymi 3001 |
. . . . . . . . 9
⊢ ¬
∅ = 1o |
77 | | iffalse 4468 |
. . . . . . . . . 10
⊢ (¬
𝑎 ∈ ran 𝐹 → if(𝑎 ∈ ran 𝐹, 1o, ∅) =
∅) |
78 | 77 | eqeq1d 2740 |
. . . . . . . . 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 3087 |
. . . . . 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 3909 |
. . . . 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 3938 |
. . 3
⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋𝐸𝑌)) → ran 𝐹 = 𝑌) |
86 | | dffo2 6692 |
. . 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 6688 |
. . . . 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 2824 |
. . . . 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 17796 |
. . . . . . . . . . . 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 17798 |
. . . . . . . . . 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 17796 |
. . . . . . . . . . . 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 17798 |
. . . . . . . . . 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 2754 |
. . . . . . . . 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 6601 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → 𝑔 Fn 𝑌) |
110 | 105 | ffnd 6601 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐹:𝑋–onto→𝑌) ∧ (𝑧 ∈ 𝑈 ∧ (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑌(Hom ‘𝐶)𝑧)))) → ℎ Fn 𝑌) |
111 | | cocan2 7164 |
. . . . . . . . . . 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 3123 |
. . . . . 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 3102 |
. . 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 17453 |
. . . 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)) |