| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . . . . 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 17908 |
. . . . . . . . . 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 495 |
. . . . . . . 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 2737 |
. . . . . . . . 9
⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸) |
| 11 | | uncfcurf.c |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ Cat) |
| 12 | 9, 10, 11, 2, 4 | curfcl 18277 |
. . . . . . . 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 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 18281 |
. . . . . 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 2737 |
. . . . . . 7
⊢
((1st ‘𝐺)‘𝑥) = ((1st ‘𝐺)‘𝑥) |
| 22 | 9, 14, 19, 3, 20, 15, 16, 21, 17 | curf11 18271 |
. . . . . 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 3202 |
. . . 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 18223 |
. . . . . . 7
⊢
((Base‘𝐶)
× (Base‘𝐷)) =
(Base‘(𝐶
×c 𝐷)) |
| 27 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝐸) =
(Base‘𝐸) |
| 28 | | relfunc 17907 |
. . . . . . . 8
⊢ Rel
((𝐶
×c 𝐷) Func 𝐸) |
| 29 | 1, 2, 7, 12 | uncfcl 18280 |
. . . . . . . 8
⊢ (𝜑 → (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) ∈ ((𝐶 ×c
𝐷) Func 𝐸)) |
| 30 | | 1st2ndbr 8067 |
. . . . . . . 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 587 |
. . . . . . 7
⊢ (𝜑 → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))) |
| 32 | 26, 27, 31 | funcf1 17911 |
. . . . . 6
⊢ (𝜑 → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸)) |
| 33 | 32 | ffnd 6737 |
. . . . 5
⊢ (𝜑 → (1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) Fn ((Base‘𝐶) × (Base‘𝐷))) |
| 34 | | 1st2ndbr 8067 |
. . . . . . . 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 587 |
. . . . . . 7
⊢ (𝜑 → (1st
‘𝐹)((𝐶 ×c
𝐷) Func 𝐸)(2nd ‘𝐹)) |
| 36 | 26, 27, 35 | funcf1 17911 |
. . . . . 6
⊢ (𝜑 → (1st
‘𝐹):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐸)) |
| 37 | 36 | ffnd 6737 |
. . . . 5
⊢ (𝜑 → (1st
‘𝐹) Fn
((Base‘𝐶) ×
(Base‘𝐷))) |
| 38 | | eqfnov2 7563 |
. . . . 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 584 |
. . . 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 257 |
. . 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 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑥 ∈ (Base‘𝐶)) |
| 45 | 44 | 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)) |
| 46 | 17 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷)) |
| 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))) → 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 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)) |
| 52 | | simprr 773 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 𝑤 ∈ (Base‘𝐷)) |
| 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))) → 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 18282 |
. . . . . . . . . 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 18271 |
. . . . . . . . . . . . . . 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 7434 |
. . . . . . . . . . . . . . 15
⊢ (𝑥(1st ‘𝐹)𝑦) = ((1st ‘𝐹)‘〈𝑥, 𝑦〉) |
| 61 | 59, 60 | eqtrdi 2793 |
. . . . . . . . . . . . . 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 18271 |
. . . . . . . . . . . . . . 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 7434 |
. . . . . . . . . . . . . . 15
⊢ (𝑥(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘〈𝑥, 𝑤〉) |
| 64 | 62, 63 | eqtrdi 2793 |
. . . . . . . . . . . . . 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 4881 |
. . . . . . . . . . . . 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 18271 |
. . . . . . . . . . . . . 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 7434 |
. . . . . . . . . . . . . 14
⊢ (𝑧(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘〈𝑧, 𝑤〉) |
| 69 | 67, 68 | eqtrdi 2793 |
. . . . . . . . . . . . 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 7449 |
. . . . . . . . . . . 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 18275 |
. . . . . . . . . . . . 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 7434 |
. . . . . . . . . . . . 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 2793 |
. . . . . . . . . . . 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 18272 |
. . . . . . . . . . . . 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 7434 |
. . . . . . . . . . . . 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 2793 |
. . . . . . . . . . . 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 7452 |
. . . . . . . . . . 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 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)) |
| 86 | | opelxpi 5722 |
. . . . . . . . . . . . . 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 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))) |
| 89 | 45, 53 | opelxpd 5724 |
. . . . . . . . . . . 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 5722 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷)) → 〈𝑧, 𝑤〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
| 91 | 90 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷))) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐷))) → 〈𝑧, 𝑤〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
| 92 | 91 | 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))) |
| 93 | 14, 48, 76, 57, 45 | catidcl 17725 |
. . . . . . . . . . . . . 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 5724 |
. . . . . . . . . . . . 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 18231 |
. . . . . . . . . . . . 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 2844 |
. . . . . . . . . . . 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 17725 |
. . . . . . . . . . . . . 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 5724 |
. . . . . . . . . . . . 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 18231 |
. . . . . . . . . . . . 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 2844 |
. . . . . . . . . . . 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 17916 |
. . . . . . . . . . 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 18232 |
. . . . . . . . . . . . . 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 6910 |
. . . . . . . . . . . . 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 7434 |
. . . . . . . . . . . . 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 2795 |
. . . . . . . . . . . 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 17727 |
. . . . . . . . . . . . 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 17726 |
. . . . . . . . . . . . 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 7449 |
. . . . . . . . . . . 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 3202 |
. . . . . . . 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 17913 |
. . . . . . . . . . 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 18231 |
. . . . . . . . . . . 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 6722 |
. . . . . . . . . . 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 232 |
. . . . . . . . . 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 6737 |
. . . . . . . . 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 17913 |
. . . . . . . . . . 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 6722 |
. . . . . . . . . . 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 232 |
. . . . . . . . . 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 6737 |
. . . . . . . . 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 7563 |
. . . . . . . . 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 584 |
. . . . . . . 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 257 |
. . . . . . 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 3202 |
. . . . . 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 3202 |
. . . . 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 7439 |
. . . . . . . . 9
⊢ (𝑣 = 〈𝑧, 𝑤〉 → (𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))𝑣) = (𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉)) |
| 132 | | oveq2 7439 |
. . . . . . . . 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 5852 |
. . . . . . 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 7438 |
. . . . . . . . 9
⊢ (𝑢 = 〈𝑥, 𝑦〉 → (𝑢(2nd ‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉) = (〈𝑥, 𝑦〉(2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〈𝑧, 𝑤〉)) |
| 136 | | oveq1 7438 |
. . . . . . . . 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 3221 |
. . . . . . 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 | bitrid 283 |
. . . . . 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 5852 |
. . . . 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 234 |
. . . 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 17914 |
. . . . 5
⊢ (𝜑 → (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))) |
| 143 | 26, 35 | funcfn2 17914 |
. . . . 5
⊢ (𝜑 → (2nd
‘𝐹) Fn
(((Base‘𝐶) ×
(Base‘𝐷)) ×
((Base‘𝐶) ×
(Base‘𝐷)))) |
| 144 | | eqfnov2 7563 |
. . . . 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 584 |
. . . 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 257 |
. . 3
⊢ (𝜑 → (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)) = (2nd
‘𝐹)) |
| 147 | 40, 146 | opeq12d 4881 |
. 2
⊢ (𝜑 → 〈(1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)), (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〉 =
〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
| 148 | | 1st2nd 8064 |
. . 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 587 |
. 2
⊢ (𝜑 → (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) = 〈(1st
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺)), (2nd
‘(〈“𝐶𝐷𝐸”〉 uncurryF
𝐺))〉) |
| 150 | | 1st2nd 8064 |
. . 3
⊢ ((Rel
((𝐶
×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
| 151 | 28, 4, 150 | sylancr 587 |
. 2
⊢ (𝜑 → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
| 152 | 147, 149,
151 | 3eqtr4d 2787 |
1
⊢ (𝜑 → (〈“𝐶𝐷𝐸”〉 uncurryF
𝐺) = 𝐹) |