Step | Hyp | Ref
| Expression |
1 | | eqid 2737 |
. . . . . . 7
⊢
(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) = (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) |
2 | | uncfcurf.d |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ Cat) |
3 | 2 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐷 ∈ Cat) |
4 | | uncfcurf.f |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
5 | | funcrcl 17369 |
. . . . . . . . . 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 499 |
. . . . . . . 8
⊢ (𝜑 → 𝐸 ∈ Cat) |
8 | 7 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐸 ∈ Cat) |
9 | | uncfcurf.g |
. . . . . . . . 9
⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) |
10 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸) |
11 | | uncfcurf.c |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ Cat) |
12 | 9, 10, 11, 2, 4 | curfcl 17740 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) |
13 | 12 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) |
14 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝐶) =
(Base‘𝐶) |
15 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝐷) =
(Base‘𝐷) |
16 | | simprl 771 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶)) |
17 | | simprr 773 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷)) |
18 | 1, 3, 8, 13, 14, 15, 16, 17 | uncf1 17744 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑦) = ((1st ‘((1st
‘𝐺)‘𝑥))‘𝑦)) |
19 | 11 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐶 ∈ Cat) |
20 | 4 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
21 | | eqid 2737 |
. . . . . . 7
⊢
((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑥) |
22 | 9, 14, 19, 3, 20, 15, 16, 21, 17 | curf11 17734 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → ((1st
‘((1st ‘𝐺)‘𝑥))‘𝑦) = (𝑥(1st ‘𝐹)𝑦)) |
23 | 18, 22 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) → (𝑥(1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦)) |
24 | 23 | ralrimivva 3112 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦)) |
25 | | eqid 2737 |
. . . . . . . 8
⊢ (𝐶 ×c
𝐷) = (𝐶 ×c 𝐷) |
26 | 25, 14, 15 | xpcbas 17685 |
. . . . . . 7
⊢
((Base‘𝐶)
× (Base‘𝐷)) =
(Base‘(𝐶
×c 𝐷)) |
27 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝐸) =
(Base‘𝐸) |
28 | | relfunc 17368 |
. . . . . . . 8
⊢ Rel
((𝐶
×c 𝐷) Func 𝐸) |
29 | 1, 2, 7, 12 | uncfcl 17743 |
. . . . . . . 8
⊢ (𝜑 → (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) ∈ ((𝐶 ×c
𝐷) Func 𝐸)) |
30 | | 1st2ndbr 7813 |
. . . . . . . 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 590 |
. . . . . . 7
⊢ (𝜑 → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))) |
32 | 26, 27, 31 | funcf1 17372 |
. . . . . 6
⊢ (𝜑 → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸)) |
33 | 32 | ffnd 6546 |
. . . . 5
⊢ (𝜑 → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷))) |
34 | | 1st2ndbr 7813 |
. . . . . . . 8
⊢ ((Rel
((𝐶
×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
35 | 28, 4, 34 | sylancr 590 |
. . . . . . 7
⊢ (𝜑 → (1st
‘𝐹)((𝐶 ×c
𝐷) Func 𝐸)(2nd ‘𝐹)) |
36 | 26, 27, 35 | funcf1 17372 |
. . . . . 6
⊢ (𝜑 → (1st
‘𝐹):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸)) |
37 | 36 | ffnd 6546 |
. . . . 5
⊢ (𝜑 → (1st
‘𝐹) Fn
((Base‘𝐶) ×
(Base‘𝐷))) |
38 | | eqfnov2 7340 |
. . . . 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))) |
39 | 33, 37, 38 | syl2anc 587 |
. . . 4
⊢ (𝜑 → ((1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) = (1st
‘𝐹) ↔
∀𝑥 ∈
(Base‘𝐶)∀𝑦 ∈ (Base‘𝐷)(𝑥(1st ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑦) = (𝑥(1st ‘𝐹)𝑦))) |
40 | 24, 39 | mpbird 260 |
. . 3
⊢ (𝜑 → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) = (1st
‘𝐹)) |
41 | 2 | ad3antrrr 730 |
. . . . . . . . . . 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) |
42 | 7 | ad3antrrr 730 |
. . . . . . . . . . 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) |
43 | 12 | ad3antrrr 730 |
. . . . . . . . . . 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))) |
44 | 16 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶)) |
45 | 44 | adantr 484 |
. . . . . . . . . . 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)) |
46 | 17 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷)) |
47 | 46 | adantr 484 |
. . . . . . . . . . 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)) |
48 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
49 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) |
50 | | simprl 771 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐶)) |
51 | 50 | adantr 484 |
. . . . . . . . . . 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)) |
52 | | simprr 773 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑤 ∈ (Base‘𝐷)) |
53 | 52 | adantr 484 |
. . . . . . . . . . 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)) |
54 | | simprl 771 |
. . . . . . . . . . 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)) |
55 | | simprr 773 |
. . . . . . . . . . 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)) |
56 | 1, 41, 42, 43, 14, 15, 45, 47, 48, 49, 51, 53, 54, 55 | uncf2 17745 |
. . . . . . . . . 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))) |
57 | 11 | ad3antrrr 730 |
. . . . . . . . . . . . . . . 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) |
58 | 4 | ad3antrrr 730 |
. . . . . . . . . . . . . . . 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)) |
59 | 9, 14, 57, 41, 58, 15, 45, 21, 47 | curf11 17734 |
. . . . . . . . . . . . . . 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)) |
60 | | df-ov 7216 |
. . . . . . . . . . . . . . 15
⊢ (𝑥(1st ‘𝐹)𝑦) = ((1st ‘𝐹)‘〈𝑥, 𝑦〉) |
61 | 59, 60 | eqtrdi 2794 |
. . . . . . . . . . . . . 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〉)) |
62 | 9, 14, 57, 41, 58, 15, 45, 21, 53 | curf11 17734 |
. . . . . . . . . . . . . . 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)) |
63 | | df-ov 7216 |
. . . . . . . . . . . . . . 15
⊢ (𝑥(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘〈𝑥, 𝑤〉) |
64 | 62, 63 | eqtrdi 2794 |
. . . . . . . . . . . . . 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〉)) |
65 | 61, 64 | opeq12d 4792 |
. . . . . . . . . . . . 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〉)〉) |
66 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
((1st ‘𝐺)‘𝑧) = ((1st ‘𝐺)‘𝑧) |
67 | 9, 14, 57, 41, 58, 15, 51, 66, 53 | curf11 17734 |
. . . . . . . . . . . . . 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)) |
68 | | df-ov 7216 |
. . . . . . . . . . . . . 14
⊢ (𝑧(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘〈𝑧, 𝑤〉) |
69 | 67, 68 | eqtrdi 2794 |
. . . . . . . . . . . . 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〉)) |
70 | 65, 69 | oveq12d 7231 |
. . . . . . . . . . . 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〉))) |
71 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(Id‘𝐷) =
(Id‘𝐷) |
72 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ ((𝑥(2nd ‘𝐺)𝑧)‘𝑓) = ((𝑥(2nd ‘𝐺)𝑧)‘𝑓) |
73 | 9, 14, 57, 41, 58, 15, 48, 71, 45, 51, 54, 72, 53 | curf2val 17738 |
. . . . . . . . . . . . 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))) |
74 | | df-ov 7216 |
. . . . . . . . . . . . 13
⊢ (𝑓(〈𝑥, 𝑤〉(2nd ‘𝐹)〈𝑧, 𝑤〉)((Id‘𝐷)‘𝑤)) = ((〈𝑥, 𝑤〉(2nd ‘𝐹)〈𝑧, 𝑤〉)‘〈𝑓, ((Id‘𝐷)‘𝑤)〉) |
75 | 73, 74 | eqtrdi 2794 |
. . . . . . . . . . . 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)〉)) |
76 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(Id‘𝐶) =
(Id‘𝐶) |
77 | 9, 14, 57, 41, 58, 15, 45, 21, 47, 49, 76, 53, 55 | curf12 17735 |
. . . . . . . . . . . . 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)) |
78 | | df-ov 7216 |
. . . . . . . . . . . . 13
⊢
(((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑤〉)𝑔) = ((〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑤〉)‘〈((Id‘𝐶)‘𝑥), 𝑔〉) |
79 | 77, 78 | eqtrdi 2794 |
. . . . . . . . . . . 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〉)) |
80 | 70, 75, 79 | oveq123d 7234 |
. . . . . . . . . . 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〉))) |
81 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (Hom
‘(𝐶
×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) |
82 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(comp‘(𝐶
×c 𝐷)) = (comp‘(𝐶 ×c 𝐷)) |
83 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(comp‘𝐸) =
(comp‘𝐸) |
84 | 35 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
85 | 84 | adantr 484 |
. . . . . . . . . . . 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)) |
86 | | opelxpi 5588 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → 〈𝑥, 𝑦〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
87 | 86 | ad2antlr 727 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 〈𝑥, 𝑦〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
88 | 87 | adantr 484 |
. . . . . . . . . . . 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))) |
89 | 45, 53 | opelxpd 5589 |
. . . . . . . . . . . 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))) |
90 | | opelxpi 5588 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → 〈𝑧, 𝑤〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
91 | 90 | adantl 485 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 〈𝑧, 𝑤〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
92 | 91 | adantr 484 |
. . . . . . . . . . . 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))) |
93 | 14, 48, 76, 57, 45 | catidcl 17185 |
. . . . . . . . . . . . . 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)) |
94 | 93, 55 | opelxpd 5589 |
. . . . . . . . . . . . 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))) |
95 | 25, 14, 15, 48, 49, 45, 47, 45, 53, 81 | xpchom2 17693 |
. . . . . . . . . . . . 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))) |
96 | 94, 95 | eleqtrrd 2841 |
. . . . . . . . . . . 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〉)) |
97 | 15, 49, 71, 41, 53 | catidcl 17185 |
. . . . . . . . . . . . . 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)) |
98 | 54, 97 | opelxpd 5589 |
. . . . . . . . . . . . 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))) |
99 | 25, 14, 15, 48, 49, 45, 53, 51, 53, 81 | xpchom2 17693 |
. . . . . . . . . . . . 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))) |
100 | 98, 99 | eleqtrrd 2841 |
. . . . . . . . . . . 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〉)) |
101 | 26, 81, 82, 83, 85, 88, 89, 92, 96, 100 | funcco 17377 |
. . . . . . . . . . 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〉))) |
102 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(comp‘𝐶) =
(comp‘𝐶) |
103 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(comp‘𝐷) =
(comp‘𝐷) |
104 | 25, 14, 15, 48, 49, 45, 47, 45, 53, 102, 103, 82, 51, 53, 93, 55, 54, 97 | xpcco2 17694 |
. . . . . . . . . . . . . 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)〉) |
105 | 104 | fveq2d 6721 |
. . . . . . . . . . . . 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)〉)) |
106 | | df-ov 7216 |
. . . . . . . . . . . . 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)〉) |
107 | 105, 106 | eqtr4di 2796 |
. . . . . . . . . . . 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))) |
108 | 14, 48, 76, 57, 45, 102, 51, 54 | catrid 17187 |
. . . . . . . . . . . . 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) |
109 | 15, 49, 71, 41, 47, 103, 53, 55 | catlid 17186 |
. . . . . . . . . . . . 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) |
110 | 108, 109 | oveq12d 7231 |
. . . . . . . . . . . 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)) |
111 | 107, 110 | eqtrd 2777 |
. . . . . . . . . . 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)) |
112 | 80, 101, 111 | 3eqtr2d 2783 |
. . . . . . . . . 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)) |
113 | 56, 112 | eqtrd 2777 |
. . . . . . . . 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)) |
114 | 113 | ralrimivva 3112 |
. . . . . . . 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)) |
115 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (Hom
‘𝐸) = (Hom
‘𝐸) |
116 | 31 | ad2antrr 726 |
. . . . . . . . . . . 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))) |
117 | 26, 81, 115, 116, 87, 91 | funcf2 17374 |
. . . . . . . . . . 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〉))) |
118 | 25, 14, 15, 48, 49, 44, 46, 50, 52, 81 | xpchom2 17693 |
. . . . . . . . . . . 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))) |
119 | 118 | feq2d 6531 |
. . . . . . . . . . 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〉)))) |
120 | 117, 119 | mpbid 235 |
. . . . . . . . . 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〉))) |
121 | 120 | ffnd 6546 |
. . . . . . . . 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))) |
122 | 26, 81, 115, 84, 87, 91 | funcf2 17374 |
. . . . . . . . . . 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〉))) |
123 | 118 | feq2d 6531 |
. . . . . . . . . . 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〉)))) |
124 | 122, 123 | mpbid 235 |
. . . . . . . . . 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〉))) |
125 | 124 | ffnd 6546 |
. . . . . . . . 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))) |
126 | | eqfnov2 7340 |
. . . . . . . . 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))) |
127 | 121, 125,
126 | syl2anc 587 |
. . . . . . . 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))) |
128 | 114, 127 | mpbird 260 |
. . . . . . 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〉)) |
129 | 128 | ralrimivva 3112 |
. . . . . 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〉)) |
130 | 129 | ralrimivva 3112 |
. . . . 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〉)) |
131 | | oveq2 7221 |
. . . . . . . . 9
⊢ (𝑣 = 〈𝑧, 𝑤〉 → (𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑣) = (𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉)) |
132 | | oveq2 7221 |
. . . . . . . . 9
⊢ (𝑣 = 〈𝑧, 𝑤〉 → (𝑢(2nd ‘𝐹)𝑣) = (𝑢(2nd ‘𝐹)〈𝑧, 𝑤〉)) |
133 | 131, 132 | eqeq12d 2753 |
. . . . . . . 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〉))) |
134 | 133 | ralxp 5710 |
. . . . . . 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〉)) |
135 | | oveq1 7220 |
. . . . . . . . 9
⊢ (𝑢 = 〈𝑥, 𝑦〉 → (𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉)) |
136 | | oveq1 7220 |
. . . . . . . . 9
⊢ (𝑢 = 〈𝑥, 𝑦〉 → (𝑢(2nd ‘𝐹)〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑧, 𝑤〉)) |
137 | 135, 136 | eqeq12d 2753 |
. . . . . . . 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〉))) |
138 | 137 | 2ralbidv 3120 |
. . . . . . 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〉))) |
139 | 134, 138 | syl5bb 286 |
. . . . . 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〉))) |
140 | 139 | ralxp 5710 |
. . . . 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〉)) |
141 | 130, 140 | sylibr 237 |
. . . 4
⊢ (𝜑 → ∀𝑢 ∈ ((Base‘𝐶) × (Base‘𝐷))∀𝑣 ∈ ((Base‘𝐶) × (Base‘𝐷))(𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑣) = (𝑢(2nd ‘𝐹)𝑣)) |
142 | 26, 31 | funcfn2 17375 |
. . . . 5
⊢ (𝜑 → (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))) |
143 | 26, 35 | funcfn2 17375 |
. . . . 5
⊢ (𝜑 → (2nd
‘𝐹) Fn
(((Base‘𝐶) ×
(Base‘𝐷)) ×
((Base‘𝐶) ×
(Base‘𝐷)))) |
144 | | eqfnov2 7340 |
. . . . 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))) |
145 | 142, 143,
144 | syl2anc 587 |
. . . 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))) |
146 | 141, 145 | mpbird 260 |
. . 3
⊢ (𝜑 → (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) = (2nd
‘𝐹)) |
147 | 40, 146 | opeq12d 4792 |
. 2
⊢ (𝜑 → 〈(1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)), (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〉 =
〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
148 | | 1st2nd 7810 |
. . 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))〉) |
149 | 28, 29, 148 | sylancr 590 |
. 2
⊢ (𝜑 → (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) = 〈(1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)), (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〉) |
150 | | 1st2nd 7810 |
. . 3
⊢ ((Rel
((𝐶
×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
151 | 28, 4, 150 | sylancr 590 |
. 2
⊢ (𝜑 → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
152 | 147, 149,
151 | 3eqtr4d 2787 |
1
⊢ (𝜑 → (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) = 𝐹) |