Step | Hyp | Ref
| Expression |
1 | | eqid 2610 |
. . . . . . 7
⊢
(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) = (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) |
2 | | uncfcurf.d |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ Cat) |
3 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat) |
4 | | uncfcurf.f |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
5 | | funcrcl 16346 |
. . . . . . . . . 10
⊢ (𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸) → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat)) |
6 | 4, 5 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat)) |
7 | 6 | simprd 478 |
. . . . . . . 8
⊢ (𝜑 → 𝐸 ∈ Cat) |
8 | 7 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐸 ∈ Cat) |
9 | | uncfcurf.g |
. . . . . . . . 9
⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) |
10 | | eqid 2610 |
. . . . . . . . 9
⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸) |
11 | | uncfcurf.c |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ Cat) |
12 | 9, 10, 11, 2, 4 | curfcl 16695 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) |
13 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) |
14 | | eqid 2610 |
. . . . . . 7
⊢
(Base‘𝐶) =
(Base‘𝐶) |
15 | | eqid 2610 |
. . . . . . 7
⊢
(Base‘𝐷) =
(Base‘𝐷) |
16 | | simprl 790 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶)) |
17 | | simprr 792 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷)) |
18 | 1, 3, 8, 13, 14, 15, 16, 17 | uncf1 16699 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑦) = ((1st ‘((1st
‘𝐺)‘𝑥))‘𝑦)) |
19 | 11 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐶 ∈ Cat) |
20 | 4 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
21 | | eqid 2610 |
. . . . . . 7
⊢
((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑥) |
22 | 9, 14, 19, 3, 20, 15, 16, 21, 17 | curf11 16689 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st
‘((1st ‘𝐺)‘𝑥))‘𝑦) = (𝑥(1st ‘𝐹)𝑦)) |
23 | 18, 22 | eqtrd 2644 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦)) |
24 | 23 | ralrimivva 2954 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦)) |
25 | | eqid 2610 |
. . . . . . . 8
⊢ (𝐶 ×c
𝐷) = (𝐶 ×c 𝐷) |
26 | 25, 14, 15 | xpcbas 16641 |
. . . . . . 7
⊢
((Base‘𝐶)
× (Base‘𝐷)) =
(Base‘(𝐶
×c 𝐷)) |
27 | | eqid 2610 |
. . . . . . 7
⊢
(Base‘𝐸) =
(Base‘𝐸) |
28 | | relfunc 16345 |
. . . . . . . 8
⊢ Rel
((𝐶
×c 𝐷) Func 𝐸) |
29 | 1, 2, 7, 12 | uncfcl 16698 |
. . . . . . . 8
⊢ (𝜑 → (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) ∈ ((𝐶 ×c
𝐷) Func 𝐸)) |
30 | | 1st2ndbr 7108 |
. . . . . . . 8
⊢ ((Rel
((𝐶
×c 𝐷) Func 𝐸) ∧ (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) ∈ ((𝐶 ×c
𝐷) Func 𝐸)) → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))) |
31 | 28, 29, 30 | sylancr 694 |
. . . . . . 7
⊢ (𝜑 → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))) |
32 | 26, 27, 31 | funcf1 16349 |
. . . . . 6
⊢ (𝜑 → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸)) |
33 | | ffn 5958 |
. . . . . 6
⊢
((1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸) → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷))) |
34 | 32, 33 | syl 17 |
. . . . 5
⊢ (𝜑 → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷))) |
35 | | 1st2ndbr 7108 |
. . . . . . . 8
⊢ ((Rel
((𝐶
×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
36 | 28, 4, 35 | sylancr 694 |
. . . . . . 7
⊢ (𝜑 → (1st
‘𝐹)((𝐶 ×c
𝐷) Func 𝐸)(2nd ‘𝐹)) |
37 | 26, 27, 36 | funcf1 16349 |
. . . . . 6
⊢ (𝜑 → (1st
‘𝐹):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸)) |
38 | | ffn 5958 |
. . . . . 6
⊢
((1st ‘𝐹):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸) → (1st ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐷))) |
39 | 37, 38 | syl 17 |
. . . . 5
⊢ (𝜑 → (1st
‘𝐹) Fn
((Base‘𝐶) ×
(Base‘𝐷))) |
40 | | eqfnov2 6665 |
. . . . 5
⊢
(((1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷)) ∧ (1st
‘𝐹) Fn
((Base‘𝐶) ×
(Base‘𝐷))) →
((1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) = (1st
‘𝐹) ↔
∀𝑥 ∈
(Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦))) |
41 | 34, 39, 40 | syl2anc 691 |
. . . 4
⊢ (𝜑 → ((1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) = (1st
‘𝐹) ↔
∀𝑥 ∈
(Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦))) |
42 | 24, 41 | mpbird 246 |
. . 3
⊢ (𝜑 → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) = (1st
‘𝐹)) |
43 | 2 | ad3antrrr 762 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat) |
44 | 7 | ad3antrrr 762 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝐸 ∈ Cat) |
45 | 12 | ad3antrrr 762 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) |
46 | 16 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶)) |
47 | 46 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑥 ∈ (Base‘𝐶)) |
48 | 17 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷)) |
49 | 48 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑦 ∈ (Base‘𝐷)) |
50 | | eqid 2610 |
. . . . . . . . . . 11
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
51 | | eqid 2610 |
. . . . . . . . . . 11
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) |
52 | | simprl 790 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐶)) |
53 | 52 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑧 ∈ (Base‘𝐶)) |
54 | | simprr 792 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑤 ∈ (Base‘𝐷)) |
55 | 54 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑤 ∈ (Base‘𝐷)) |
56 | | simprl 790 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧)) |
57 | | simprr 792 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤)) |
58 | 1, 43, 44, 45, 14, 15, 47, 49, 50, 51, 53, 55, 56, 57 | uncf2 16700 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (𝑓(〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉)𝑔) = ((((𝑥(2nd ‘𝐺)𝑧)‘𝑓)‘𝑤)(〈((1st
‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st
‘𝐺)‘𝑥))‘𝑤)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑧))‘𝑤))((𝑦(2nd ‘((1st
‘𝐺)‘𝑥))𝑤)‘𝑔))) |
59 | 11 | ad3antrrr 762 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat) |
60 | 4 | ad3antrrr 762 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
61 | 9, 14, 59, 43, 60, 15, 47, 21, 49 | curf11 16689 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑥))‘𝑦) = (𝑥(1st ‘𝐹)𝑦)) |
62 | | df-ov 6552 |
. . . . . . . . . . . . . . 15
⊢ (𝑥(1st ‘𝐹)𝑦) = ((1st ‘𝐹)‘〈𝑥, 𝑦〉) |
63 | 61, 62 | syl6eq 2660 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑥))‘𝑦) = ((1st ‘𝐹)‘〈𝑥, 𝑦〉)) |
64 | 9, 14, 59, 43, 60, 15, 47, 21, 55 | curf11 16689 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑥))‘𝑤) = (𝑥(1st ‘𝐹)𝑤)) |
65 | | df-ov 6552 |
. . . . . . . . . . . . . . 15
⊢ (𝑥(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘〈𝑥, 𝑤〉) |
66 | 64, 65 | syl6eq 2660 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑥))‘𝑤) = ((1st ‘𝐹)‘〈𝑥, 𝑤〉)) |
67 | 63, 66 | opeq12d 4348 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 〈((1st
‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st
‘𝐺)‘𝑥))‘𝑤)〉 = 〈((1st ‘𝐹)‘〈𝑥, 𝑦〉), ((1st ‘𝐹)‘〈𝑥, 𝑤〉)〉) |
68 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢
((1st ‘𝐺)‘𝑧) = ((1st ‘𝐺)‘𝑧) |
69 | 9, 14, 59, 43, 60, 15, 53, 68, 55 | curf11 16689 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑧))‘𝑤) = (𝑧(1st ‘𝐹)𝑤)) |
70 | | df-ov 6552 |
. . . . . . . . . . . . . 14
⊢ (𝑧(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘〈𝑧, 𝑤〉) |
71 | 69, 70 | syl6eq 2660 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑧))‘𝑤) = ((1st ‘𝐹)‘〈𝑧, 𝑤〉)) |
72 | 67, 71 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (〈((1st
‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st
‘𝐺)‘𝑥))‘𝑤)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑧))‘𝑤)) = (〈((1st ‘𝐹)‘〈𝑥, 𝑦〉), ((1st ‘𝐹)‘〈𝑥, 𝑤〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑧, 𝑤〉))) |
73 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢
(Id‘𝐷) =
(Id‘𝐷) |
74 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢ ((𝑥(2nd ‘𝐺)𝑧)‘𝑓) = ((𝑥(2nd ‘𝐺)𝑧)‘𝑓) |
75 | 9, 14, 59, 43, 60, 15, 50, 73, 47, 53, 56, 74, 55 | curf2val 16693 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (((𝑥(2nd ‘𝐺)𝑧)‘𝑓)‘𝑤) = (𝑓(〈𝑥, 𝑤〉(2nd ‘𝐹)〈𝑧, 𝑤〉)((Id‘𝐷)‘𝑤))) |
76 | | df-ov 6552 |
. . . . . . . . . . . . 13
⊢ (𝑓(〈𝑥, 𝑤〉(2nd ‘𝐹)〈𝑧, 𝑤〉)((Id‘𝐷)‘𝑤)) = ((〈𝑥, 𝑤〉(2nd ‘𝐹)〈𝑧, 𝑤〉)‘〈𝑓, ((Id‘𝐷)‘𝑤)〉) |
77 | 75, 76 | syl6eq 2660 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (((𝑥(2nd ‘𝐺)𝑧)‘𝑓)‘𝑤) = ((〈𝑥, 𝑤〉(2nd ‘𝐹)〈𝑧, 𝑤〉)‘〈𝑓, ((Id‘𝐷)‘𝑤)〉)) |
78 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢
(Id‘𝐶) =
(Id‘𝐶) |
79 | 9, 14, 59, 43, 60, 15, 47, 21, 49, 51, 78, 55, 57 | curf12 16690 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘((1st
‘𝐺)‘𝑥))𝑤)‘𝑔) = (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑤〉)𝑔)) |
80 | | df-ov 6552 |
. . . . . . . . . . . . 13
⊢
(((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑤〉)𝑔) = ((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑤〉)‘〈((Id‘𝐶)‘𝑥), 𝑔〉) |
81 | 79, 80 | syl6eq 2660 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘((1st
‘𝐺)‘𝑥))𝑤)‘𝑔) = ((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑤〉)‘〈((Id‘𝐶)‘𝑥), 𝑔〉)) |
82 | 72, 77, 81 | oveq123d 6570 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((((𝑥(2nd ‘𝐺)𝑧)‘𝑓)‘𝑤)(〈((1st
‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st
‘𝐺)‘𝑥))‘𝑤)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑧))‘𝑤))((𝑦(2nd ‘((1st
‘𝐺)‘𝑥))𝑤)‘𝑔)) = (((〈𝑥, 𝑤〉(2nd ‘𝐹)〈𝑧, 𝑤〉)‘〈𝑓, ((Id‘𝐷)‘𝑤)〉)(〈((1st ‘𝐹)‘〈𝑥, 𝑦〉), ((1st ‘𝐹)‘〈𝑥, 𝑤〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑧, 𝑤〉))((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑤〉)‘〈((Id‘𝐶)‘𝑥), 𝑔〉))) |
83 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (Hom
‘(𝐶
×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) |
84 | | eqid 2610 |
. . . . . . . . . . . 12
⊢
(comp‘(𝐶
×c 𝐷)) = (comp‘(𝐶 ×c 𝐷)) |
85 | | eqid 2610 |
. . . . . . . . . . . 12
⊢
(comp‘𝐸) =
(comp‘𝐸) |
86 | 36 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
87 | 86 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
88 | | opelxpi 5072 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → 〈𝑥, 𝑦〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
89 | 88 | ad2antlr 759 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 〈𝑥, 𝑦〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
90 | 89 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 〈𝑥, 𝑦〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
91 | | opelxpi 5072 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → 〈𝑥, 𝑤〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
92 | 47, 55, 91 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 〈𝑥, 𝑤〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
93 | | opelxpi 5072 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → 〈𝑧, 𝑤〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
94 | 93 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 〈𝑧, 𝑤〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
95 | 94 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 〈𝑧, 𝑤〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
96 | 14, 50, 78, 59, 47 | catidcl 16166 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥)) |
97 | | opelxpi 5072 |
. . . . . . . . . . . . . 14
⊢
((((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤)) → 〈((Id‘𝐶)‘𝑥), 𝑔〉 ∈ ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑤))) |
98 | 96, 57, 97 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 〈((Id‘𝐶)‘𝑥), 𝑔〉 ∈ ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑤))) |
99 | 25, 14, 15, 50, 51, 47, 49, 47, 55, 83 | xpchom2 16649 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑥, 𝑤〉) = ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑤))) |
100 | 98, 99 | eleqtrrd 2691 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 〈((Id‘𝐶)‘𝑥), 𝑔〉 ∈ (〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑥, 𝑤〉)) |
101 | 15, 51, 73, 43, 55 | catidcl 16166 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((Id‘𝐷)‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤)) |
102 | | opelxpi 5072 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ((Id‘𝐷)‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤)) → 〈𝑓, ((Id‘𝐷)‘𝑤)〉 ∈ ((𝑥(Hom ‘𝐶)𝑧) × (𝑤(Hom ‘𝐷)𝑤))) |
103 | 56, 101, 102 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 〈𝑓, ((Id‘𝐷)‘𝑤)〉 ∈ ((𝑥(Hom ‘𝐶)𝑧) × (𝑤(Hom ‘𝐷)𝑤))) |
104 | 25, 14, 15, 50, 51, 47, 55, 53, 55, 83 | xpchom2 16649 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (〈𝑥, 𝑤〉(Hom ‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉) = ((𝑥(Hom ‘𝐶)𝑧) × (𝑤(Hom ‘𝐷)𝑤))) |
105 | 103, 104 | eleqtrrd 2691 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → 〈𝑓, ((Id‘𝐷)‘𝑤)〉 ∈ (〈𝑥, 𝑤〉(Hom ‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉)) |
106 | 26, 83, 84, 85, 87, 90, 92, 95, 100, 105 | funcco 16354 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)‘(〈𝑓, ((Id‘𝐷)‘𝑤)〉(〈〈𝑥, 𝑦〉, 〈𝑥, 𝑤〉〉(comp‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉)〈((Id‘𝐶)‘𝑥), 𝑔〉)) = (((〈𝑥, 𝑤〉(2nd ‘𝐹)〈𝑧, 𝑤〉)‘〈𝑓, ((Id‘𝐷)‘𝑤)〉)(〈((1st ‘𝐹)‘〈𝑥, 𝑦〉), ((1st ‘𝐹)‘〈𝑥, 𝑤〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑧, 𝑤〉))((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑤〉)‘〈((Id‘𝐶)‘𝑥), 𝑔〉))) |
107 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢
(comp‘𝐶) =
(comp‘𝐶) |
108 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢
(comp‘𝐷) =
(comp‘𝐷) |
109 | 25, 14, 15, 50, 51, 47, 49, 47, 55, 107, 108, 84, 53, 55, 96, 57, 56, 101 | xpcco2 16650 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (〈𝑓, ((Id‘𝐷)‘𝑤)〉(〈〈𝑥, 𝑦〉, 〈𝑥, 𝑤〉〉(comp‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉)〈((Id‘𝐶)‘𝑥), 𝑔〉) = 〈(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥)), (((Id‘𝐷)‘𝑤)(〈𝑦, 𝑤〉(comp‘𝐷)𝑤)𝑔)〉) |
110 | 109 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)‘(〈𝑓, ((Id‘𝐷)‘𝑤)〉(〈〈𝑥, 𝑦〉, 〈𝑥, 𝑤〉〉(comp‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉)〈((Id‘𝐶)‘𝑥), 𝑔〉)) = ((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)‘〈(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥)), (((Id‘𝐷)‘𝑤)(〈𝑦, 𝑤〉(comp‘𝐷)𝑤)𝑔)〉)) |
111 | | df-ov 6552 |
. . . . . . . . . . . . 13
⊢ ((𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥))(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)(((Id‘𝐷)‘𝑤)(〈𝑦, 𝑤〉(comp‘𝐷)𝑤)𝑔)) = ((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)‘〈(𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥)), (((Id‘𝐷)‘𝑤)(〈𝑦, 𝑤〉(comp‘𝐷)𝑤)𝑔)〉) |
112 | 110, 111 | syl6eqr 2662 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)‘(〈𝑓, ((Id‘𝐷)‘𝑤)〉(〈〈𝑥, 𝑦〉, 〈𝑥, 𝑤〉〉(comp‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉)〈((Id‘𝐶)‘𝑥), 𝑔〉)) = ((𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥))(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)(((Id‘𝐷)‘𝑤)(〈𝑦, 𝑤〉(comp‘𝐷)𝑤)𝑔))) |
113 | 14, 50, 78, 59, 47, 107, 53, 56 | catrid 16168 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥)) = 𝑓) |
114 | 15, 51, 73, 43, 49, 108, 55, 57 | catlid 16167 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (((Id‘𝐷)‘𝑤)(〈𝑦, 𝑤〉(comp‘𝐷)𝑤)𝑔) = 𝑔) |
115 | 113, 114 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((𝑓(〈𝑥, 𝑥〉(comp‘𝐶)𝑧)((Id‘𝐶)‘𝑥))(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)(((Id‘𝐷)‘𝑤)(〈𝑦, 𝑤〉(comp‘𝐷)𝑤)𝑔)) = (𝑓(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)𝑔)) |
116 | 112, 115 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)‘(〈𝑓, ((Id‘𝐷)‘𝑤)〉(〈〈𝑥, 𝑦〉, 〈𝑥, 𝑤〉〉(comp‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉)〈((Id‘𝐶)‘𝑥), 𝑔〉)) = (𝑓(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)𝑔)) |
117 | 82, 106, 116 | 3eqtr2d 2650 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → ((((𝑥(2nd ‘𝐺)𝑧)‘𝑓)‘𝑤)(〈((1st
‘((1st ‘𝐺)‘𝑥))‘𝑦), ((1st ‘((1st
‘𝐺)‘𝑥))‘𝑤)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑧))‘𝑤))((𝑦(2nd ‘((1st
‘𝐺)‘𝑥))𝑤)‘𝑔)) = (𝑓(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)𝑔)) |
118 | 58, 117 | eqtrd 2644 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤))) → (𝑓(〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉)𝑔) = (𝑓(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)𝑔)) |
119 | 118 | ralrimivva 2954 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤)(𝑓(〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉)𝑔) = (𝑓(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)𝑔)) |
120 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (Hom
‘𝐸) = (Hom
‘𝐸) |
121 | 31 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))) |
122 | 26, 83, 120, 121, 89, 94 | funcf2 16351 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉):(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉)⟶(((1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))‘〈𝑥, 𝑦〉)(Hom ‘𝐸)((1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))‘〈𝑧, 𝑤〉))) |
123 | 25, 14, 15, 50, 51, 46, 48, 52, 54, 83 | xpchom2 16649 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉) = ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))) |
124 | 123 | feq2d 5944 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ((〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉):(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉)⟶(((1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))‘〈𝑥, 𝑦〉)(Hom ‘𝐸)((1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))‘〈𝑧, 𝑤〉)) ↔ (〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))‘〈𝑥, 𝑦〉)(Hom ‘𝐸)((1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))‘〈𝑧, 𝑤〉)))) |
125 | 122, 124 | mpbid 221 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))‘〈𝑥, 𝑦〉)(Hom ‘𝐸)((1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))‘〈𝑧, 𝑤〉))) |
126 | | ffn 5958 |
. . . . . . . . . 10
⊢
((〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))‘〈𝑥, 𝑦〉)(Hom ‘𝐸)((1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))‘〈𝑧, 𝑤〉)) → (〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))) |
127 | 125, 126 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))) |
128 | 26, 83, 120, 86, 89, 94 | funcf2 16351 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉):(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉)⟶(((1st ‘𝐹)‘〈𝑥, 𝑦〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑧, 𝑤〉))) |
129 | 123 | feq2d 5944 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉):(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑧, 𝑤〉)⟶(((1st ‘𝐹)‘〈𝑥, 𝑦〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑧, 𝑤〉)) ↔ (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st ‘𝐹)‘〈𝑥, 𝑦〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑧, 𝑤〉)))) |
130 | 128, 129 | mpbid 221 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st ‘𝐹)‘〈𝑥, 𝑦〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑧, 𝑤〉))) |
131 | | ffn 5958 |
. . . . . . . . . 10
⊢
((〈𝑥, 𝑦〉(2nd
‘𝐹)〈𝑧, 𝑤〉):((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))⟶(((1st ‘𝐹)‘〈𝑥, 𝑦〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑧, 𝑤〉)) → (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))) |
132 | 130, 131 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))) |
133 | | eqfnov2 6665 |
. . . . . . . . 9
⊢
(((〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤)) ∧ (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉) Fn ((𝑥(Hom ‘𝐶)𝑧) × (𝑦(Hom ‘𝐷)𝑤))) → ((〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤)(𝑓(〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉)𝑔) = (𝑓(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)𝑔))) |
134 | 127, 132,
133 | syl2anc 691 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → ((〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑧)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑤)(𝑓(〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉)𝑔) = (𝑓(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)𝑔))) |
135 | 119, 134 | mpbird 246 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)) |
136 | 135 | ralrimivva 2954 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)) |
137 | 136 | ralrimivva 2954 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)) |
138 | | oveq2 6557 |
. . . . . . . . 9
⊢ (𝑣 = 〈𝑧, 𝑤〉 → (𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑣) = (𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉)) |
139 | | oveq2 6557 |
. . . . . . . . 9
⊢ (𝑣 = 〈𝑧, 𝑤〉 → (𝑢(2nd ‘𝐹)𝑣) = (𝑢(2nd ‘𝐹)〈𝑧, 𝑤〉)) |
140 | 138, 139 | eqeq12d 2625 |
. . . . . . . 8
⊢ (𝑣 = 〈𝑧, 𝑤〉 → ((𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣) ↔ (𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (𝑢(2nd ‘𝐹)〈𝑧, 𝑤〉))) |
141 | 140 | ralxp 5185 |
. . . . . . 7
⊢
(∀𝑣 ∈
((Base‘𝐶) ×
(Base‘𝐷))(𝑢(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (𝑢(2nd ‘𝐹)〈𝑧, 𝑤〉)) |
142 | | oveq1 6556 |
. . . . . . . . 9
⊢ (𝑢 = 〈𝑥, 𝑦〉 → (𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉)) |
143 | | oveq1 6556 |
. . . . . . . . 9
⊢ (𝑢 = 〈𝑥, 𝑦〉 → (𝑢(2nd ‘𝐹)〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)) |
144 | 142, 143 | eqeq12d 2625 |
. . . . . . . 8
⊢ (𝑢 = 〈𝑥, 𝑦〉 → ((𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (𝑢(2nd ‘𝐹)〈𝑧, 𝑤〉) ↔ (〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉))) |
145 | 144 | 2ralbidv 2972 |
. . . . . . 7
⊢ (𝑢 = 〈𝑥, 𝑦〉 → (∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (𝑢(2nd ‘𝐹)〈𝑧, 𝑤〉) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉))) |
146 | 141, 145 | syl5bb 271 |
. . . . . 6
⊢ (𝑢 = 〈𝑥, 𝑦〉 → (∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉))) |
147 | 146 | ralxp 5185 |
. . . . 5
⊢
(∀𝑢 ∈
((Base‘𝐶) ×
(Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐷)(〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)) |
148 | 137, 147 | sylibr 223 |
. . . 4
⊢ (𝜑 → ∀𝑢 ∈ ((Base‘𝐶) × (Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣)) |
149 | 26, 31 | funcfn2 16352 |
. . . . 5
⊢ (𝜑 → (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))) |
150 | 26, 36 | funcfn2 16352 |
. . . . 5
⊢ (𝜑 → (2nd
‘𝐹) Fn
(((Base‘𝐶) ×
(Base‘𝐷)) ×
((Base‘𝐶) ×
(Base‘𝐷)))) |
151 | | eqfnov2 6665 |
. . . . 5
⊢
(((2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))) ∧ (2nd
‘𝐹) Fn
(((Base‘𝐶) ×
(Base‘𝐷)) ×
((Base‘𝐶) ×
(Base‘𝐷)))) →
((2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) = (2nd
‘𝐹) ↔
∀𝑢 ∈
((Base‘𝐶) ×
(Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣))) |
152 | 149, 150,
151 | syl2anc 691 |
. . . 4
⊢ (𝜑 → ((2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) = (2nd
‘𝐹) ↔
∀𝑢 ∈
((Base‘𝐶) ×
(Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣))) |
153 | 148, 152 | mpbird 246 |
. . 3
⊢ (𝜑 → (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) = (2nd
‘𝐹)) |
154 | 42, 153 | opeq12d 4348 |
. 2
⊢ (𝜑 → 〈(1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)), (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〉 =
〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
155 | | 1st2nd 7105 |
. . 3
⊢ ((Rel
((𝐶
×c 𝐷) Func 𝐸) ∧ (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) ∈ ((𝐶 ×c
𝐷) Func 𝐸)) → (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) = 〈(1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)), (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〉) |
156 | 28, 29, 155 | sylancr 694 |
. 2
⊢ (𝜑 → (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) = 〈(1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)), (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〉) |
157 | | 1st2nd 7105 |
. . 3
⊢ ((Rel
((𝐶
×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
158 | 28, 4, 157 | sylancr 694 |
. 2
⊢ (𝜑 → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
159 | 154, 156,
158 | 3eqtr4d 2654 |
1
⊢ (𝜑 → (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) = 𝐹) |