Step | Hyp | Ref
| Expression |
1 | | df-ov 6653 |
. . . . . . . . . . 11
⊢ (𝑥(1st ‘(𝐶
2ndF 𝐷))𝑦) = ((1st ‘(𝐶
2ndF 𝐷))‘〈𝑥, 𝑦〉) |
2 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ (𝐶 ×c
𝐷) = (𝐶 ×c 𝐷) |
3 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝐶) =
(Base‘𝐶) |
4 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝐷) =
(Base‘𝐷) |
5 | 2, 3, 4 | xpcbas 16818 |
. . . . . . . . . . . . 13
⊢
((Base‘𝐶)
× (Base‘𝐷)) =
(Base‘(𝐶
×c 𝐷)) |
6 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ (Hom
‘(𝐶
×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) |
7 | | curf2ndf.c |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐶 ∈ Cat) |
8 | 7 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat) |
9 | | curf2ndf.d |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐷 ∈ Cat) |
10 | 9 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat) |
11 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ (𝐶
2ndF 𝐷) = (𝐶 2ndF 𝐷) |
12 | | opelxpi 5148 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → 〈𝑥, 𝑦〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
13 | 12 | adantll 750 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 〈𝑥, 𝑦〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
14 | 2, 5, 6, 8, 10, 11, 13 | 2ndf1 16835 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘(𝐶
2ndF 𝐷))‘〈𝑥, 𝑦〉) = (2nd ‘〈𝑥, 𝑦〉)) |
15 | | vex 3203 |
. . . . . . . . . . . . 13
⊢ 𝑥 ∈ V |
16 | | vex 3203 |
. . . . . . . . . . . . 13
⊢ 𝑦 ∈ V |
17 | 15, 16 | op2nd 7177 |
. . . . . . . . . . . 12
⊢
(2nd ‘〈𝑥, 𝑦〉) = 𝑦 |
18 | 14, 17 | syl6eq 2672 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘(𝐶
2ndF 𝐷))‘〈𝑥, 𝑦〉) = 𝑦) |
19 | 1, 18 | syl5eq 2668 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦) = 𝑦) |
20 | 19 | mpteq2dva 4744 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)) = (𝑦 ∈ (Base‘𝐷) ↦ 𝑦)) |
21 | | mptresid 5456 |
. . . . . . . . 9
⊢ (𝑦 ∈ (Base‘𝐷) ↦ 𝑦) = ( I ↾ (Base‘𝐷)) |
22 | 20, 21 | syl6eq 2672 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)) = ( I ↾ (Base‘𝐷))) |
23 | | df-ov 6653 |
. . . . . . . . . . . . . . 15
⊢
(((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓) = ((〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)‘〈((Id‘𝐶)‘𝑥), 𝑓〉) |
24 | 8 | ad2antrr 762 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat) |
25 | 10 | ad2antrr 762 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat) |
26 | 13 | ad2antrr 762 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 〈𝑥, 𝑦〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
27 | | simp-4r 807 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑥 ∈ (Base‘𝐶)) |
28 | | simplr 792 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧 ∈ (Base‘𝐷)) |
29 | | opelxpi 5148 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑥, 𝑧〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
30 | 27, 28, 29 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 〈𝑥, 𝑧〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
31 | 2, 5, 6, 24, 25, 11, 26, 30 | 2ndf2 16836 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉) = (2nd ↾
(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c
𝐷))〈𝑥, 𝑧〉))) |
32 | 31 | fveq1d 6193 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)‘〈((Id‘𝐶)‘𝑥), 𝑓〉) = ((2nd ↾
(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c
𝐷))〈𝑥, 𝑧〉))‘〈((Id‘𝐶)‘𝑥), 𝑓〉)) |
33 | 23, 32 | syl5eq 2668 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓) = ((2nd ↾ (〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑥, 𝑧〉))‘〈((Id‘𝐶)‘𝑥), 𝑓〉)) |
34 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
35 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(Id‘𝐶) =
(Id‘𝐶) |
36 | 7 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat) |
37 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶)) |
38 | 3, 34, 35, 36, 37 | catidcl 16343 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥)) |
39 | 38 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥)) |
40 | | simpr 477 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) |
41 | | opelxpi 5148 |
. . . . . . . . . . . . . . . . . 18
⊢
((((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 〈((Id‘𝐶)‘𝑥), 𝑓〉 ∈ ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑧))) |
42 | 39, 40, 41 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 〈((Id‘𝐶)‘𝑥), 𝑓〉 ∈ ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑧))) |
43 | | eqid 2622 |
. . . . . . . . . . . . . . . . . 18
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) |
44 | | simpllr 799 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦 ∈ (Base‘𝐷)) |
45 | 2, 3, 4, 34, 43, 27, 44, 27, 28, 6 | xpchom2 16826 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑥, 𝑧〉) = ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑧))) |
46 | 42, 45 | eleqtrrd 2704 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 〈((Id‘𝐶)‘𝑥), 𝑓〉 ∈ (〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑥, 𝑧〉)) |
47 | | fvres 6207 |
. . . . . . . . . . . . . . . 16
⊢
(〈((Id‘𝐶)‘𝑥), 𝑓〉 ∈ (〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑥, 𝑧〉) → ((2nd ↾
(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c
𝐷))〈𝑥, 𝑧〉))‘〈((Id‘𝐶)‘𝑥), 𝑓〉) = (2nd
‘〈((Id‘𝐶)‘𝑥), 𝑓〉)) |
48 | 46, 47 | syl 17 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ↾
(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c
𝐷))〈𝑥, 𝑧〉))‘〈((Id‘𝐶)‘𝑥), 𝑓〉) = (2nd
‘〈((Id‘𝐶)‘𝑥), 𝑓〉)) |
49 | | fvex 6201 |
. . . . . . . . . . . . . . . 16
⊢
((Id‘𝐶)‘𝑥) ∈ V |
50 | | vex 3203 |
. . . . . . . . . . . . . . . 16
⊢ 𝑓 ∈ V |
51 | 49, 50 | op2nd 7177 |
. . . . . . . . . . . . . . 15
⊢
(2nd ‘〈((Id‘𝐶)‘𝑥), 𝑓〉) = 𝑓 |
52 | 48, 51 | syl6eq 2672 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ↾
(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c
𝐷))〈𝑥, 𝑧〉))‘〈((Id‘𝐶)‘𝑥), 𝑓〉) = 𝑓) |
53 | 33, 52 | eqtrd 2656 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓) = 𝑓) |
54 | 53 | mpteq2dva 4744 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓)) = (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ 𝑓)) |
55 | | mptresid 5456 |
. . . . . . . . . . . 12
⊢ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ 𝑓) = ( I ↾ (𝑦(Hom ‘𝐷)𝑧)) |
56 | 54, 55 | syl6eq 2672 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓)) = ( I ↾ (𝑦(Hom ‘𝐷)𝑧))) |
57 | 56 | 3impa 1259 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓)) = ( I ↾ (𝑦(Hom ‘𝐷)𝑧))) |
58 | 57 | mpt2eq3dva 6719 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓))) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ ( I ↾ (𝑦(Hom ‘𝐷)𝑧)))) |
59 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑢 = 〈𝑦, 𝑧〉 → ((Hom ‘𝐷)‘𝑢) = ((Hom ‘𝐷)‘〈𝑦, 𝑧〉)) |
60 | | df-ov 6653 |
. . . . . . . . . . . 12
⊢ (𝑦(Hom ‘𝐷)𝑧) = ((Hom ‘𝐷)‘〈𝑦, 𝑧〉) |
61 | 59, 60 | syl6eqr 2674 |
. . . . . . . . . . 11
⊢ (𝑢 = 〈𝑦, 𝑧〉 → ((Hom ‘𝐷)‘𝑢) = (𝑦(Hom ‘𝐷)𝑧)) |
62 | 61 | reseq2d 5396 |
. . . . . . . . . 10
⊢ (𝑢 = 〈𝑦, 𝑧〉 → ( I ↾ ((Hom ‘𝐷)‘𝑢)) = ( I ↾ (𝑦(Hom ‘𝐷)𝑧))) |
63 | 62 | mpt2mpt 6752 |
. . . . . . . . 9
⊢ (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom
‘𝐷)‘𝑢))) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ ( I ↾ (𝑦(Hom ‘𝐷)𝑧))) |
64 | 58, 63 | syl6eqr 2674 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓))) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom ‘𝐷)‘𝑢)))) |
65 | 22, 64 | opeq12d 4410 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 〈(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓)))〉 = 〈( I ↾
(Base‘𝐷)), (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom
‘𝐷)‘𝑢)))〉) |
66 | | eqid 2622 |
. . . . . . . 8
⊢
(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)) = (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) |
67 | 9 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat) |
68 | 2, 7, 9, 11 | 2ndfcl 16838 |
. . . . . . . . 9
⊢ (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷)) |
69 | 68 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷)) |
70 | | eqid 2622 |
. . . . . . . 8
⊢
((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))‘𝑥) = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))‘𝑥) |
71 | 66, 3, 36, 67, 69, 4, 37, 70, 43, 35 | curf1 16865 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥) = 〈(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓)))〉) |
72 | | eqid 2622 |
. . . . . . . 8
⊢
(idfunc‘𝐷) = (idfunc‘𝐷) |
73 | 72, 4, 67, 43 | idfuval 16536 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) →
(idfunc‘𝐷) = 〈( I ↾ (Base‘𝐷)), (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom ‘𝐷)‘𝑢)))〉) |
74 | 65, 71, 73 | 3eqtr4d 2666 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥) = (idfunc‘𝐷)) |
75 | | eqid 2622 |
. . . . . . 7
⊢ (𝑄Δfunc𝐶) = (𝑄Δfunc𝐶) |
76 | | curf2ndf.q |
. . . . . . . . 9
⊢ 𝑄 = (𝐷 FuncCat 𝐷) |
77 | 76, 9, 9 | fuccat 16630 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ∈ Cat) |
78 | 77 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑄 ∈ Cat) |
79 | 76 | fucbas 16620 |
. . . . . . 7
⊢ (𝐷 Func 𝐷) = (Base‘𝑄) |
80 | 72 | idfucl 16541 |
. . . . . . . . 9
⊢ (𝐷 ∈ Cat →
(idfunc‘𝐷) ∈ (𝐷 Func 𝐷)) |
81 | 9, 80 | syl 17 |
. . . . . . . 8
⊢ (𝜑 →
(idfunc‘𝐷) ∈ (𝐷 Func 𝐷)) |
82 | 81 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) →
(idfunc‘𝐷) ∈ (𝐷 Func 𝐷)) |
83 | | eqid 2622 |
. . . . . . 7
⊢
((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) = ((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) |
84 | 75, 78, 36, 79, 82, 83, 3, 37 | diag11 16883 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑥) = (idfunc‘𝐷)) |
85 | 74, 84 | eqtr4d 2659 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥) = ((1st ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑥)) |
86 | 85 | mpteq2dva 4744 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑥))) |
87 | | relfunc 16522 |
. . . . . . 7
⊢ Rel
(𝐶 Func 𝑄) |
88 | 66, 76, 7, 9, 68 | curfcl 16872 |
. . . . . . 7
⊢ (𝜑 → (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) ∈ (𝐶 Func 𝑄)) |
89 | | 1st2ndbr 7217 |
. . . . . . 7
⊢ ((Rel
(𝐶 Func 𝑄) ∧ (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) ∈ (𝐶 Func 𝑄)) → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))(𝐶 Func 𝑄)(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))) |
90 | 87, 88, 89 | sylancr 695 |
. . . . . 6
⊢ (𝜑 → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))(𝐶 Func 𝑄)(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))) |
91 | 3, 79, 90 | funcf1 16526 |
. . . . 5
⊢ (𝜑 → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))):(Base‘𝐶)⟶(𝐷 Func 𝐷)) |
92 | 91 | feqmptd 6249 |
. . . 4
⊢ (𝜑 → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥))) |
93 | 75, 77, 7, 79, 81, 83 | diag1cl 16882 |
. . . . . . 7
⊢ (𝜑 → ((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) ∈ (𝐶 Func 𝑄)) |
94 | | 1st2ndbr 7217 |
. . . . . . 7
⊢ ((Rel
(𝐶 Func 𝑄) ∧ ((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) ∈ (𝐶 Func 𝑄)) → (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))(𝐶 Func 𝑄)(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))) |
95 | 87, 93, 94 | sylancr 695 |
. . . . . 6
⊢ (𝜑 → (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))(𝐶 Func 𝑄)(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))) |
96 | 3, 79, 95 | funcf1 16526 |
. . . . 5
⊢ (𝜑 → (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))):(Base‘𝐶)⟶(𝐷 Func 𝐷)) |
97 | 96 | feqmptd 6249 |
. . . 4
⊢ (𝜑 → (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑥))) |
98 | 86, 92, 97 | 3eqtr4d 2666 |
. . 3
⊢ (𝜑 → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) = (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))) |
99 | 9 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat) |
100 | 72, 4, 99 | idfu1st 16539 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st
‘(idfunc‘𝐷)) = ( I ↾ (Base‘𝐷))) |
101 | 100 | coeq2d 5284 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((Id‘𝐷) ∘ (1st
‘(idfunc‘𝐷))) = ((Id‘𝐷) ∘ ( I ↾ (Base‘𝐷)))) |
102 | | eqid 2622 |
. . . . . . . . . . 11
⊢
(Id‘𝑄) =
(Id‘𝑄) |
103 | | eqid 2622 |
. . . . . . . . . . 11
⊢
(Id‘𝐷) =
(Id‘𝐷) |
104 | 81 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) →
(idfunc‘𝐷) ∈ (𝐷 Func 𝐷)) |
105 | 76, 102, 103, 104 | fucid 16631 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((Id‘𝑄)‘(idfunc‘𝐷)) = ((Id‘𝐷) ∘ (1st
‘(idfunc‘𝐷)))) |
106 | 4, 103 | cidfn 16340 |
. . . . . . . . . . . . . 14
⊢ (𝐷 ∈ Cat →
(Id‘𝐷) Fn
(Base‘𝐷)) |
107 | 99, 106 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Id‘𝐷) Fn (Base‘𝐷)) |
108 | | dffn2 6047 |
. . . . . . . . . . . . 13
⊢
((Id‘𝐷) Fn
(Base‘𝐷) ↔
(Id‘𝐷):(Base‘𝐷)⟶V) |
109 | 107, 108 | sylib 208 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Id‘𝐷):(Base‘𝐷)⟶V) |
110 | 109 | feqmptd 6249 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Id‘𝐷) = (𝑧 ∈ (Base‘𝐷) ↦ ((Id‘𝐷)‘𝑧))) |
111 | | fcoi1 6078 |
. . . . . . . . . . . 12
⊢
((Id‘𝐷):(Base‘𝐷)⟶V → ((Id‘𝐷) ∘ ( I ↾
(Base‘𝐷))) =
(Id‘𝐷)) |
112 | 109, 111 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((Id‘𝐷) ∘ ( I ↾ (Base‘𝐷))) = (Id‘𝐷)) |
113 | 7 | ad2antrr 762 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐶 ∈ Cat) |
114 | 113 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat) |
115 | 99 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat) |
116 | | simplrl 800 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶)) |
117 | 116, 29 | sylan 488 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑥, 𝑧〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
118 | | simplrr 801 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶)) |
119 | | opelxpi 5148 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑦, 𝑧〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
120 | 118, 119 | sylan 488 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑦, 𝑧〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
121 | 2, 5, 6, 114, 115, 11, 117, 120 | 2ndf2 16836 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉) = (2nd ↾
(〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c
𝐷))〈𝑦, 𝑧〉))) |
122 | 121 | oveqd 6667 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧)) = (𝑓(2nd ↾ (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉))((Id‘𝐷)‘𝑧))) |
123 | | df-ov 6653 |
. . . . . . . . . . . . . . 15
⊢ (𝑓(2nd ↾
(〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c
𝐷))〈𝑦, 𝑧〉))((Id‘𝐷)‘𝑧)) = ((2nd ↾ (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉))‘〈𝑓, ((Id‘𝐷)‘𝑧)〉) |
124 | | simplr 792 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) |
125 | | simpr 477 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑧 ∈ (Base‘𝐷)) |
126 | 4, 43, 103, 115, 125 | catidcl 16343 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((Id‘𝐷)‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧)) |
127 | | opelxpi 5148 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((Id‘𝐷)‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧)) → 〈𝑓, ((Id‘𝐷)‘𝑧)〉 ∈ ((𝑥(Hom ‘𝐶)𝑦) × (𝑧(Hom ‘𝐷)𝑧))) |
128 | 124, 126,
127 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑓, ((Id‘𝐷)‘𝑧)〉 ∈ ((𝑥(Hom ‘𝐶)𝑦) × (𝑧(Hom ‘𝐷)𝑧))) |
129 | 116 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶)) |
130 | 118 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐶)) |
131 | 2, 3, 4, 34, 43, 129, 125, 130, 125, 6 | xpchom2 16826 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉) = ((𝑥(Hom ‘𝐶)𝑦) × (𝑧(Hom ‘𝐷)𝑧))) |
132 | 128, 131 | eleqtrrd 2704 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑓, ((Id‘𝐷)‘𝑧)〉 ∈ (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉)) |
133 | | fvres 6207 |
. . . . . . . . . . . . . . . 16
⊢
(〈𝑓,
((Id‘𝐷)‘𝑧)〉 ∈ (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉) → ((2nd ↾
(〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c
𝐷))〈𝑦, 𝑧〉))‘〈𝑓, ((Id‘𝐷)‘𝑧)〉) = (2nd ‘〈𝑓, ((Id‘𝐷)‘𝑧)〉)) |
134 | 132, 133 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((2nd ↾
(〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c
𝐷))〈𝑦, 𝑧〉))‘〈𝑓, ((Id‘𝐷)‘𝑧)〉) = (2nd ‘〈𝑓, ((Id‘𝐷)‘𝑧)〉)) |
135 | 123, 134 | syl5eq 2668 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(2nd ↾ (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉))((Id‘𝐷)‘𝑧)) = (2nd ‘〈𝑓, ((Id‘𝐷)‘𝑧)〉)) |
136 | | fvex 6201 |
. . . . . . . . . . . . . . 15
⊢
((Id‘𝐷)‘𝑧) ∈ V |
137 | 50, 136 | op2nd 7177 |
. . . . . . . . . . . . . 14
⊢
(2nd ‘〈𝑓, ((Id‘𝐷)‘𝑧)〉) = ((Id‘𝐷)‘𝑧) |
138 | 135, 137 | syl6eq 2672 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(2nd ↾ (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉))((Id‘𝐷)‘𝑧)) = ((Id‘𝐷)‘𝑧)) |
139 | 122, 138 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧)) = ((Id‘𝐷)‘𝑧)) |
140 | 139 | mpteq2dva 4744 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧))) = (𝑧 ∈ (Base‘𝐷) ↦ ((Id‘𝐷)‘𝑧))) |
141 | 110, 112,
140 | 3eqtr4rd 2667 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧))) = ((Id‘𝐷) ∘ ( I ↾ (Base‘𝐷)))) |
142 | 101, 105,
141 | 3eqtr4rd 2667 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧))) = ((Id‘𝑄)‘(idfunc‘𝐷))) |
143 | 68 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷)) |
144 | | simpr 477 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) |
145 | | eqid 2622 |
. . . . . . . . . 10
⊢ ((𝑥(2nd
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))𝑦)‘𝑓) = ((𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)‘𝑓) |
146 | 66, 3, 113, 99, 143, 4, 34, 103, 116, 118, 144, 145 | curf2 16869 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)‘𝑓) = (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧)))) |
147 | 77 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑄 ∈ Cat) |
148 | 75, 147, 113, 79, 104, 83, 3, 116, 34, 102, 118, 144 | diag12 16884 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)‘𝑓) = ((Id‘𝑄)‘(idfunc‘𝐷))) |
149 | 142, 146,
148 | 3eqtr4d 2666 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)‘𝑓) = ((𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)‘𝑓)) |
150 | 149 | mpteq2dva 4744 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)‘𝑓)) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)‘𝑓))) |
151 | | eqid 2622 |
. . . . . . . . . 10
⊢ (𝐷 Nat 𝐷) = (𝐷 Nat 𝐷) |
152 | 76, 151 | fuchom 16621 |
. . . . . . . . 9
⊢ (𝐷 Nat 𝐷) = (Hom ‘𝑄) |
153 | 90 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))(𝐶 Func 𝑄)(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))) |
154 | | simprl 794 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) |
155 | | simprr 796 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) |
156 | 3, 34, 152, 153, 154, 155 | funcf2 16528 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥)(𝐷 Nat 𝐷)((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))‘𝑦))) |
157 | 156 | feqmptd 6249 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)‘𝑓))) |
158 | 95 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))(𝐶 Func 𝑄)(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))) |
159 | 3, 34, 152, 158, 154, 155 | funcf2 16528 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑥)(𝐷 Nat 𝐷)((1st ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑦))) |
160 | 159 | feqmptd 6249 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)‘𝑓))) |
161 | 150, 157,
160 | 3eqtr4d 2666 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦) = (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)) |
162 | 161 | 3impb 1260 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦) = (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)) |
163 | 162 | mpt2eq3dva 6719 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦))) |
164 | 3, 90 | funcfn2 16529 |
. . . . 5
⊢ (𝜑 → (2nd
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶))) |
165 | | fnov 6768 |
. . . . 5
⊢
((2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦))) |
166 | 164, 165 | sylib 208 |
. . . 4
⊢ (𝜑 → (2nd
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦))) |
167 | 3, 95 | funcfn2 16529 |
. . . . 5
⊢ (𝜑 → (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶))) |
168 | | fnov 6768 |
. . . . 5
⊢
((2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦))) |
169 | 167, 168 | sylib 208 |
. . . 4
⊢ (𝜑 → (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦))) |
170 | 163, 166,
169 | 3eqtr4d 2666 |
. . 3
⊢ (𝜑 → (2nd
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) = (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))) |
171 | 98, 170 | opeq12d 4410 |
. 2
⊢ (𝜑 → 〈(1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))), (2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))〉 = 〈(1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))), (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))〉) |
172 | | 1st2nd 7214 |
. . 3
⊢ ((Rel
(𝐶 Func 𝑄) ∧ (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) ∈ (𝐶 Func 𝑄)) → (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) = 〈(1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))), (2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))〉) |
173 | 87, 88, 172 | sylancr 695 |
. 2
⊢ (𝜑 → (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) = 〈(1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))), (2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))〉) |
174 | | 1st2nd 7214 |
. . 3
⊢ ((Rel
(𝐶 Func 𝑄) ∧ ((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) ∈ (𝐶 Func 𝑄)) → ((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) = 〈(1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))), (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))〉) |
175 | 87, 93, 174 | sylancr 695 |
. 2
⊢ (𝜑 → ((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) = 〈(1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))), (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))〉) |
176 | 171, 173,
175 | 3eqtr4d 2666 |
1
⊢ (𝜑 → (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) = ((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) |