| Step | Hyp | Ref
| Expression |
| 1 | | curf2.g |
. . . 4
⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) |
| 2 | | curf2.a |
. . . 4
⊢ 𝐴 = (Base‘𝐶) |
| 3 | | curf2.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ Cat) |
| 4 | | curf2.d |
. . . 4
⊢ (𝜑 → 𝐷 ∈ Cat) |
| 5 | | curf2.f |
. . . 4
⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
| 6 | | curf2.b |
. . . 4
⊢ 𝐵 = (Base‘𝐷) |
| 7 | | curf2.h |
. . . 4
⊢ 𝐻 = (Hom ‘𝐶) |
| 8 | | curf2.i |
. . . 4
⊢ 𝐼 = (Id‘𝐷) |
| 9 | | curf2.x |
. . . 4
⊢ (𝜑 → 𝑋 ∈ 𝐴) |
| 10 | | curf2.y |
. . . 4
⊢ (𝜑 → 𝑌 ∈ 𝐴) |
| 11 | | curf2.k |
. . . 4
⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) |
| 12 | | curf2.l |
. . . 4
⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾) |
| 13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12 | curf2 18274 |
. . 3
⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧)))) |
| 14 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝐶 ×c
𝐷) = (𝐶 ×c 𝐷) |
| 15 | 14, 2, 6 | xpcbas 18223 |
. . . . . . . . 9
⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷)) |
| 16 | | eqid 2737 |
. . . . . . . . 9
⊢ (Hom
‘(𝐶
×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) |
| 17 | | eqid 2737 |
. . . . . . . . 9
⊢ (Hom
‘𝐸) = (Hom
‘𝐸) |
| 18 | | relfunc 17907 |
. . . . . . . . . . 11
⊢ Rel
((𝐶
×c 𝐷) Func 𝐸) |
| 19 | | 1st2ndbr 8067 |
. . . . . . . . . . 11
⊢ ((Rel
((𝐶
×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
| 20 | 18, 5, 19 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘𝐹)((𝐶 ×c
𝐷) Func 𝐸)(2nd ‘𝐹)) |
| 21 | 20 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
| 22 | | opelxpi 5722 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 〈𝑋, 𝑧〉 ∈ (𝐴 × 𝐵)) |
| 23 | 9, 22 | sylan 580 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 〈𝑋, 𝑧〉 ∈ (𝐴 × 𝐵)) |
| 24 | | opelxpi 5722 |
. . . . . . . . . 10
⊢ ((𝑌 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 〈𝑌, 𝑧〉 ∈ (𝐴 × 𝐵)) |
| 25 | 10, 24 | sylan 580 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 〈𝑌, 𝑧〉 ∈ (𝐴 × 𝐵)) |
| 26 | 15, 16, 17, 21, 23, 25 | funcf2 17913 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉):(〈𝑋, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑌, 𝑧〉)⟶(((1st ‘𝐹)‘〈𝑋, 𝑧〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑧〉))) |
| 27 | | eqid 2737 |
. . . . . . . . . 10
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) |
| 28 | 9 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐴) |
| 29 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵) |
| 30 | 10 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑌 ∈ 𝐴) |
| 31 | 14, 2, 6, 7, 27, 28, 29, 30, 29, 16 | xpchom2 18231 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (〈𝑋, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑌, 𝑧〉) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧))) |
| 32 | 31 | feq2d 6722 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉):(〈𝑋, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑌, 𝑧〉)⟶(((1st ‘𝐹)‘〈𝑋, 𝑧〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑧〉)) ↔ (〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉):((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧))⟶(((1st ‘𝐹)‘〈𝑋, 𝑧〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑧〉)))) |
| 33 | 26, 32 | mpbid 232 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉):((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧))⟶(((1st ‘𝐹)‘〈𝑋, 𝑧〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑧〉))) |
| 34 | 11 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐾 ∈ (𝑋𝐻𝑌)) |
| 35 | 4 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐷 ∈ Cat) |
| 36 | 6, 27, 8, 35, 29 | catidcl 17725 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐼‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧)) |
| 37 | 33, 34, 36 | fovcdmd 7605 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧)) ∈ (((1st ‘𝐹)‘〈𝑋, 𝑧〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑧〉))) |
| 38 | 3 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐶 ∈ Cat) |
| 39 | 5 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
| 40 | | eqid 2737 |
. . . . . . . . 9
⊢
((1st ‘𝐺)‘𝑋) = ((1st ‘𝐺)‘𝑋) |
| 41 | 1, 2, 38, 35, 39, 6, 28, 40, 29 | curf11 18271 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧) = (𝑋(1st ‘𝐹)𝑧)) |
| 42 | | df-ov 7434 |
. . . . . . . 8
⊢ (𝑋(1st ‘𝐹)𝑧) = ((1st ‘𝐹)‘〈𝑋, 𝑧〉) |
| 43 | 41, 42 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧) = ((1st ‘𝐹)‘〈𝑋, 𝑧〉)) |
| 44 | | eqid 2737 |
. . . . . . . . 9
⊢
((1st ‘𝐺)‘𝑌) = ((1st ‘𝐺)‘𝑌) |
| 45 | 1, 2, 38, 35, 39, 6, 30, 44, 29 | curf11 18271 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1st
‘((1st ‘𝐺)‘𝑌))‘𝑧) = (𝑌(1st ‘𝐹)𝑧)) |
| 46 | | df-ov 7434 |
. . . . . . . 8
⊢ (𝑌(1st ‘𝐹)𝑧) = ((1st ‘𝐹)‘〈𝑌, 𝑧〉) |
| 47 | 45, 46 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1st
‘((1st ‘𝐺)‘𝑌))‘𝑧) = ((1st ‘𝐹)‘〈𝑌, 𝑧〉)) |
| 48 | 43, 47 | oveq12d 7449 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧)) = (((1st ‘𝐹)‘〈𝑋, 𝑧〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑧〉))) |
| 49 | 37, 48 | eleqtrrd 2844 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧)) ∈ (((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧))) |
| 50 | 49 | ralrimiva 3146 |
. . . 4
⊢ (𝜑 → ∀𝑧 ∈ 𝐵 (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧)) ∈ (((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧))) |
| 51 | 6 | fvexi 6920 |
. . . . 5
⊢ 𝐵 ∈ V |
| 52 | | mptelixpg 8975 |
. . . . 5
⊢ (𝐵 ∈ V → ((𝑧 ∈ 𝐵 ↦ (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧))) ∈ X𝑧 ∈ 𝐵 (((1st ‘((1st
‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧)) ∈ (((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧)))) |
| 53 | 51, 52 | ax-mp 5 |
. . . 4
⊢ ((𝑧 ∈ 𝐵 ↦ (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧))) ∈ X𝑧 ∈ 𝐵 (((1st ‘((1st
‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧)) ∈ (((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧))) |
| 54 | 50, 53 | sylibr 234 |
. . 3
⊢ (𝜑 → (𝑧 ∈ 𝐵 ↦ (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧))) ∈ X𝑧 ∈ 𝐵 (((1st ‘((1st
‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧))) |
| 55 | 13, 54 | eqeltrd 2841 |
. 2
⊢ (𝜑 → 𝐿 ∈ X𝑧 ∈ 𝐵 (((1st ‘((1st
‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧))) |
| 56 | | eqid 2737 |
. . . . . . . . . 10
⊢
(Id‘𝐶) =
(Id‘𝐶) |
| 57 | 3 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat) |
| 58 | 9 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋 ∈ 𝐴) |
| 59 | | eqid 2737 |
. . . . . . . . . 10
⊢
(comp‘𝐶) =
(comp‘𝐶) |
| 60 | 10 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑌 ∈ 𝐴) |
| 61 | 11 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐾 ∈ (𝑋𝐻𝑌)) |
| 62 | 2, 7, 56, 57, 58, 59, 60, 61 | catrid 17727 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐾(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐾) |
| 63 | 2, 7, 56, 57, 58, 59, 60, 61 | catlid 17726 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐾) = 𝐾) |
| 64 | 62, 63 | eqtr4d 2780 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐾(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐾)) |
| 65 | 4 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat) |
| 66 | | simpr1 1195 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧 ∈ 𝐵) |
| 67 | | eqid 2737 |
. . . . . . . . . 10
⊢
(comp‘𝐷) =
(comp‘𝐷) |
| 68 | | simpr2 1196 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤 ∈ 𝐵) |
| 69 | | simpr3 1197 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)) |
| 70 | 6, 27, 8, 65, 66, 67, 68, 69 | catlid 17726 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼‘𝑤)(〈𝑧, 𝑤〉(comp‘𝐷)𝑤)𝑓) = 𝑓) |
| 71 | 6, 27, 8, 65, 66, 67, 68, 69 | catrid 17727 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝑓(〈𝑧, 𝑧〉(comp‘𝐷)𝑤)(𝐼‘𝑧)) = 𝑓) |
| 72 | 70, 71 | eqtr4d 2780 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐼‘𝑤)(〈𝑧, 𝑤〉(comp‘𝐷)𝑤)𝑓) = (𝑓(〈𝑧, 𝑧〉(comp‘𝐷)𝑤)(𝐼‘𝑧))) |
| 73 | 64, 72 | opeq12d 4881 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈(𝐾(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)), ((𝐼‘𝑤)(〈𝑧, 𝑤〉(comp‘𝐷)𝑤)𝑓)〉 = 〈(((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐾), (𝑓(〈𝑧, 𝑧〉(comp‘𝐷)𝑤)(𝐼‘𝑧))〉) |
| 74 | | eqid 2737 |
. . . . . . . 8
⊢
(comp‘(𝐶
×c 𝐷)) = (comp‘(𝐶 ×c 𝐷)) |
| 75 | 2, 7, 56, 57, 58 | catidcl 17725 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋)) |
| 76 | 6, 27, 8, 65, 68 | catidcl 17725 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼‘𝑤) ∈ (𝑤(Hom ‘𝐷)𝑤)) |
| 77 | 14, 2, 6, 7, 27, 58, 66, 58, 68, 59, 67, 74, 60, 68, 75, 69, 61, 76 | xpcco2 18232 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈𝐾, (𝐼‘𝑤)〉(〈〈𝑋, 𝑧〉, 〈𝑋, 𝑤〉〉(comp‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉)〈((Id‘𝐶)‘𝑋), 𝑓〉) = 〈(𝐾(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)), ((𝐼‘𝑤)(〈𝑧, 𝑤〉(comp‘𝐷)𝑤)𝑓)〉) |
| 78 | 36 | 3ad2antr1 1189 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐼‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧)) |
| 79 | 2, 7, 56, 57, 60 | catidcl 17725 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐻𝑌)) |
| 80 | 14, 2, 6, 7, 27, 58, 66, 60, 66, 59, 67, 74, 60, 68, 61, 78, 79, 69 | xpcco2 18232 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈((Id‘𝐶)‘𝑌), 𝑓〉(〈〈𝑋, 𝑧〉, 〈𝑌, 𝑧〉〉(comp‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉)〈𝐾, (𝐼‘𝑧)〉) = 〈(((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐾), (𝑓(〈𝑧, 𝑧〉(comp‘𝐷)𝑤)(𝐼‘𝑧))〉) |
| 81 | 73, 77, 80 | 3eqtr4d 2787 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈𝐾, (𝐼‘𝑤)〉(〈〈𝑋, 𝑧〉, 〈𝑋, 𝑤〉〉(comp‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉)〈((Id‘𝐶)‘𝑋), 𝑓〉) = (〈((Id‘𝐶)‘𝑌), 𝑓〉(〈〈𝑋, 𝑧〉, 〈𝑌, 𝑧〉〉(comp‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉)〈𝐾, (𝐼‘𝑧)〉)) |
| 82 | 81 | fveq2d 6910 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘(〈𝐾, (𝐼‘𝑤)〉(〈〈𝑋, 𝑧〉, 〈𝑋, 𝑤〉〉(comp‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉)〈((Id‘𝐶)‘𝑋), 𝑓〉)) = ((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘(〈((Id‘𝐶)‘𝑌), 𝑓〉(〈〈𝑋, 𝑧〉, 〈𝑌, 𝑧〉〉(comp‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉)〈𝐾, (𝐼‘𝑧)〉))) |
| 83 | | eqid 2737 |
. . . . . 6
⊢
(comp‘𝐸) =
(comp‘𝐸) |
| 84 | 20 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
| 85 | 23 | 3ad2antr1 1189 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈𝑋, 𝑧〉 ∈ (𝐴 × 𝐵)) |
| 86 | 58, 68 | opelxpd 5724 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈𝑋, 𝑤〉 ∈ (𝐴 × 𝐵)) |
| 87 | 60, 68 | opelxpd 5724 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈𝑌, 𝑤〉 ∈ (𝐴 × 𝐵)) |
| 88 | 75, 69 | opelxpd 5724 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈((Id‘𝐶)‘𝑋), 𝑓〉 ∈ ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤))) |
| 89 | 14, 2, 6, 7, 27, 58, 66, 58, 68, 16 | xpchom2 18231 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈𝑋, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑤〉) = ((𝑋𝐻𝑋) × (𝑧(Hom ‘𝐷)𝑤))) |
| 90 | 88, 89 | eleqtrrd 2844 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈((Id‘𝐶)‘𝑋), 𝑓〉 ∈ (〈𝑋, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑤〉)) |
| 91 | 61, 76 | opelxpd 5724 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈𝐾, (𝐼‘𝑤)〉 ∈ ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤))) |
| 92 | 14, 2, 6, 7, 27, 58, 68, 60, 68, 16 | xpchom2 18231 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈𝑋, 𝑤〉(Hom ‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉) = ((𝑋𝐻𝑌) × (𝑤(Hom ‘𝐷)𝑤))) |
| 93 | 91, 92 | eleqtrrd 2844 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈𝐾, (𝐼‘𝑤)〉 ∈ (〈𝑋, 𝑤〉(Hom ‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉)) |
| 94 | 15, 16, 74, 83, 84, 85, 86, 87, 90, 93 | funcco 17916 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘(〈𝐾, (𝐼‘𝑤)〉(〈〈𝑋, 𝑧〉, 〈𝑋, 𝑤〉〉(comp‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉)〈((Id‘𝐶)‘𝑋), 𝑓〉)) = (((〈𝑋, 𝑤〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘〈𝐾, (𝐼‘𝑤)〉)(〈((1st ‘𝐹)‘〈𝑋, 𝑧〉), ((1st ‘𝐹)‘〈𝑋, 𝑤〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑤〉))((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), 𝑓〉))) |
| 95 | 25 | 3ad2antr1 1189 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈𝑌, 𝑧〉 ∈ (𝐴 × 𝐵)) |
| 96 | 61, 78 | opelxpd 5724 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈𝐾, (𝐼‘𝑧)〉 ∈ ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧))) |
| 97 | 14, 2, 6, 7, 27, 58, 66, 60, 66, 16 | xpchom2 18231 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈𝑋, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑌, 𝑧〉) = ((𝑋𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑧))) |
| 98 | 96, 97 | eleqtrrd 2844 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈𝐾, (𝐼‘𝑧)〉 ∈ (〈𝑋, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑌, 𝑧〉)) |
| 99 | 79, 69 | opelxpd 5724 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈((Id‘𝐶)‘𝑌), 𝑓〉 ∈ ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤))) |
| 100 | 14, 2, 6, 7, 27, 60, 66, 60, 68, 16 | xpchom2 18231 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈𝑌, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉) = ((𝑌𝐻𝑌) × (𝑧(Hom ‘𝐷)𝑤))) |
| 101 | 99, 100 | eleqtrrd 2844 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈((Id‘𝐶)‘𝑌), 𝑓〉 ∈ (〈𝑌, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉)) |
| 102 | 15, 16, 74, 83, 84, 85, 95, 87, 98, 101 | funcco 17916 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘(〈((Id‘𝐶)‘𝑌), 𝑓〉(〈〈𝑋, 𝑧〉, 〈𝑌, 𝑧〉〉(comp‘(𝐶 ×c 𝐷))〈𝑌, 𝑤〉)〈𝐾, (𝐼‘𝑧)〉)) = (((〈𝑌, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘〈((Id‘𝐶)‘𝑌), 𝑓〉)(〈((1st ‘𝐹)‘〈𝑋, 𝑧〉), ((1st ‘𝐹)‘〈𝑌, 𝑧〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑤〉))((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)‘〈𝐾, (𝐼‘𝑧)〉))) |
| 103 | 82, 94, 102 | 3eqtr3d 2785 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((〈𝑋, 𝑤〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘〈𝐾, (𝐼‘𝑤)〉)(〈((1st ‘𝐹)‘〈𝑋, 𝑧〉), ((1st ‘𝐹)‘〈𝑋, 𝑤〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑤〉))((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), 𝑓〉)) = (((〈𝑌, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘〈((Id‘𝐶)‘𝑌), 𝑓〉)(〈((1st ‘𝐹)‘〈𝑋, 𝑧〉), ((1st ‘𝐹)‘〈𝑌, 𝑧〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑤〉))((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)‘〈𝐾, (𝐼‘𝑧)〉))) |
| 104 | 5 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
| 105 | 1, 2, 57, 65, 104, 6, 58, 40, 66 | curf11 18271 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧) = (𝑋(1st ‘𝐹)𝑧)) |
| 106 | 105, 42 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧) = ((1st ‘𝐹)‘〈𝑋, 𝑧〉)) |
| 107 | 1, 2, 57, 65, 104, 6, 58, 40, 68 | curf11 18271 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑋))‘𝑤) = (𝑋(1st ‘𝐹)𝑤)) |
| 108 | | df-ov 7434 |
. . . . . . . 8
⊢ (𝑋(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘〈𝑋, 𝑤〉) |
| 109 | 107, 108 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑋))‘𝑤) = ((1st ‘𝐹)‘〈𝑋, 𝑤〉)) |
| 110 | 106, 109 | opeq12d 4881 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑋))‘𝑤)〉 = 〈((1st ‘𝐹)‘〈𝑋, 𝑧〉), ((1st ‘𝐹)‘〈𝑋, 𝑤〉)〉) |
| 111 | 1, 2, 57, 65, 104, 6, 60, 44, 68 | curf11 18271 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑌))‘𝑤) = (𝑌(1st ‘𝐹)𝑤)) |
| 112 | | df-ov 7434 |
. . . . . . 7
⊢ (𝑌(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘〈𝑌, 𝑤〉) |
| 113 | 111, 112 | eqtrdi 2793 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑌))‘𝑤) = ((1st ‘𝐹)‘〈𝑌, 𝑤〉)) |
| 114 | 110, 113 | oveq12d 7449 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑋))‘𝑤)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑤)) = (〈((1st ‘𝐹)‘〈𝑋, 𝑧〉), ((1st ‘𝐹)‘〈𝑋, 𝑤〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑤〉))) |
| 115 | 1, 2, 57, 65, 104, 6, 7, 8, 58,
60, 61, 12, 68 | curf2val 18275 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑤) = (𝐾(〈𝑋, 𝑤〉(2nd ‘𝐹)〈𝑌, 𝑤〉)(𝐼‘𝑤))) |
| 116 | | df-ov 7434 |
. . . . . 6
⊢ (𝐾(〈𝑋, 𝑤〉(2nd ‘𝐹)〈𝑌, 𝑤〉)(𝐼‘𝑤)) = ((〈𝑋, 𝑤〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘〈𝐾, (𝐼‘𝑤)〉) |
| 117 | 115, 116 | eqtrdi 2793 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑤) = ((〈𝑋, 𝑤〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘〈𝐾, (𝐼‘𝑤)〉)) |
| 118 | 1, 2, 57, 65, 104, 6, 58, 40, 66, 27, 56, 68, 69 | curf12 18272 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st
‘𝐺)‘𝑋))𝑤)‘𝑓) = (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)𝑓)) |
| 119 | | df-ov 7434 |
. . . . . 6
⊢
(((Id‘𝐶)‘𝑋)(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)𝑓) = ((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), 𝑓〉) |
| 120 | 118, 119 | eqtrdi 2793 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st
‘𝐺)‘𝑋))𝑤)‘𝑓) = ((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), 𝑓〉)) |
| 121 | 114, 117,
120 | oveq123d 7452 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐿‘𝑤)(〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑋))‘𝑤)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑤))((𝑧(2nd ‘((1st
‘𝐺)‘𝑋))𝑤)‘𝑓)) = (((〈𝑋, 𝑤〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘〈𝐾, (𝐼‘𝑤)〉)(〈((1st ‘𝐹)‘〈𝑋, 𝑧〉), ((1st ‘𝐹)‘〈𝑋, 𝑤〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑤〉))((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), 𝑓〉))) |
| 122 | 1, 2, 57, 65, 104, 6, 60, 44, 66 | curf11 18271 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑌))‘𝑧) = (𝑌(1st ‘𝐹)𝑧)) |
| 123 | 122, 46 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st
‘((1st ‘𝐺)‘𝑌))‘𝑧) = ((1st ‘𝐹)‘〈𝑌, 𝑧〉)) |
| 124 | 106, 123 | opeq12d 4881 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧)〉 = 〈((1st ‘𝐹)‘〈𝑋, 𝑧〉), ((1st ‘𝐹)‘〈𝑌, 𝑧〉)〉) |
| 125 | 124, 113 | oveq12d 7449 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑤)) = (〈((1st ‘𝐹)‘〈𝑋, 𝑧〉), ((1st ‘𝐹)‘〈𝑌, 𝑧〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑤〉))) |
| 126 | 1, 2, 57, 65, 104, 6, 60, 44, 66, 27, 56, 68, 69 | curf12 18272 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st
‘𝐺)‘𝑌))𝑤)‘𝑓) = (((Id‘𝐶)‘𝑌)(〈𝑌, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑤〉)𝑓)) |
| 127 | | df-ov 7434 |
. . . . . 6
⊢
(((Id‘𝐶)‘𝑌)(〈𝑌, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑤〉)𝑓) = ((〈𝑌, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘〈((Id‘𝐶)‘𝑌), 𝑓〉) |
| 128 | 126, 127 | eqtrdi 2793 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘((1st
‘𝐺)‘𝑌))𝑤)‘𝑓) = ((〈𝑌, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘〈((Id‘𝐶)‘𝑌), 𝑓〉)) |
| 129 | 1, 2, 57, 65, 104, 6, 7, 8, 58,
60, 61, 12, 66 | curf2val 18275 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑧) = (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧))) |
| 130 | | df-ov 7434 |
. . . . . 6
⊢ (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧)) = ((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)‘〈𝐾, (𝐼‘𝑧)〉) |
| 131 | 129, 130 | eqtrdi 2793 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (𝐿‘𝑧) = ((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)‘〈𝐾, (𝐼‘𝑧)〉)) |
| 132 | 125, 128,
131 | oveq123d 7452 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((𝑧(2nd ‘((1st
‘𝐺)‘𝑌))𝑤)‘𝑓)(〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑤))(𝐿‘𝑧)) = (((〈𝑌, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑤〉)‘〈((Id‘𝐶)‘𝑌), 𝑓〉)(〈((1st ‘𝐹)‘〈𝑋, 𝑧〉), ((1st ‘𝐹)‘〈𝑌, 𝑧〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑌, 𝑤〉))((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)‘〈𝐾, (𝐼‘𝑧)〉))) |
| 133 | 103, 121,
132 | 3eqtr4d 2787 |
. . 3
⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝐿‘𝑤)(〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑋))‘𝑤)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑤))((𝑧(2nd ‘((1st
‘𝐺)‘𝑋))𝑤)‘𝑓)) = (((𝑧(2nd ‘((1st
‘𝐺)‘𝑌))𝑤)‘𝑓)(〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑤))(𝐿‘𝑧))) |
| 134 | 133 | ralrimivvva 3205 |
. 2
⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)((𝐿‘𝑤)(〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑋))‘𝑤)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑤))((𝑧(2nd ‘((1st
‘𝐺)‘𝑋))𝑤)‘𝑓)) = (((𝑧(2nd ‘((1st
‘𝐺)‘𝑌))𝑤)‘𝑓)(〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑤))(𝐿‘𝑧))) |
| 135 | | curf2.n |
. . 3
⊢ 𝑁 = (𝐷 Nat 𝐸) |
| 136 | 1, 2, 3, 4, 5, 6, 9, 40 | curf1cl 18273 |
. . 3
⊢ (𝜑 → ((1st
‘𝐺)‘𝑋) ∈ (𝐷 Func 𝐸)) |
| 137 | 1, 2, 3, 4, 5, 6, 10, 44 | curf1cl 18273 |
. . 3
⊢ (𝜑 → ((1st
‘𝐺)‘𝑌) ∈ (𝐷 Func 𝐸)) |
| 138 | 135, 6, 27, 17, 83, 136, 137 | isnat2 17996 |
. 2
⊢ (𝜑 → (𝐿 ∈ (((1st ‘𝐺)‘𝑋)𝑁((1st ‘𝐺)‘𝑌)) ↔ (𝐿 ∈ X𝑧 ∈ 𝐵 (((1st ‘((1st
‘𝐺)‘𝑋))‘𝑧)(Hom ‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧)) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑧(Hom ‘𝐷)𝑤)((𝐿‘𝑤)(〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑋))‘𝑤)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑤))((𝑧(2nd ‘((1st
‘𝐺)‘𝑋))𝑤)‘𝑓)) = (((𝑧(2nd ‘((1st
‘𝐺)‘𝑌))𝑤)‘𝑓)(〈((1st
‘((1st ‘𝐺)‘𝑋))‘𝑧), ((1st ‘((1st
‘𝐺)‘𝑌))‘𝑧)〉(comp‘𝐸)((1st ‘((1st
‘𝐺)‘𝑌))‘𝑤))(𝐿‘𝑧))))) |
| 139 | 55, 134, 138 | mpbir2and 713 |
1
⊢ (𝜑 → 𝐿 ∈ (((1st ‘𝐺)‘𝑋)𝑁((1st ‘𝐺)‘𝑌))) |