| Step | Hyp | Ref
| Expression |
| 1 | | curfval.g |
. . . 4
⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) |
| 2 | | curfval.a |
. . . 4
⊢ 𝐴 = (Base‘𝐶) |
| 3 | | curfval.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ Cat) |
| 4 | | curfval.d |
. . . 4
⊢ (𝜑 → 𝐷 ∈ Cat) |
| 5 | | curfval.f |
. . . 4
⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
| 6 | | curfval.b |
. . . 4
⊢ 𝐵 = (Base‘𝐷) |
| 7 | | curf1.x |
. . . 4
⊢ (𝜑 → 𝑋 ∈ 𝐴) |
| 8 | | curf1.k |
. . . 4
⊢ 𝐾 = ((1st ‘𝐺)‘𝑋) |
| 9 | | eqid 2737 |
. . . 4
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) |
| 10 | | eqid 2737 |
. . . 4
⊢
(Id‘𝐶) =
(Id‘𝐶) |
| 11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | curf1 18270 |
. . 3
⊢ (𝜑 → 𝐾 = 〈(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)))〉) |
| 12 | 6 | fvexi 6920 |
. . . . . . 7
⊢ 𝐵 ∈ V |
| 13 | 12 | mptex 7243 |
. . . . . 6
⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)) ∈ V |
| 14 | 12, 12 | mpoex 8104 |
. . . . . 6
⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔))) ∈ V |
| 15 | 13, 14 | op1std 8024 |
. . . . 5
⊢ (𝐾 = 〈(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)))〉 → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦))) |
| 16 | 11, 15 | syl 17 |
. . . 4
⊢ (𝜑 → (1st
‘𝐾) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦))) |
| 17 | 13, 14 | op2ndd 8025 |
. . . . 5
⊢ (𝐾 = 〈(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)))〉 → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)))) |
| 18 | 11, 17 | syl 17 |
. . . 4
⊢ (𝜑 → (2nd
‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)))) |
| 19 | 16, 18 | opeq12d 4881 |
. . 3
⊢ (𝜑 → 〈(1st
‘𝐾), (2nd
‘𝐾)〉 =
〈(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)))〉) |
| 20 | 11, 19 | eqtr4d 2780 |
. 2
⊢ (𝜑 → 𝐾 = 〈(1st ‘𝐾), (2nd ‘𝐾)〉) |
| 21 | | eqid 2737 |
. . . 4
⊢
(Base‘𝐸) =
(Base‘𝐸) |
| 22 | | eqid 2737 |
. . . 4
⊢ (Hom
‘𝐸) = (Hom
‘𝐸) |
| 23 | | eqid 2737 |
. . . 4
⊢
(Id‘𝐷) =
(Id‘𝐷) |
| 24 | | eqid 2737 |
. . . 4
⊢
(Id‘𝐸) =
(Id‘𝐸) |
| 25 | | eqid 2737 |
. . . 4
⊢
(comp‘𝐷) =
(comp‘𝐷) |
| 26 | | eqid 2737 |
. . . 4
⊢
(comp‘𝐸) =
(comp‘𝐸) |
| 27 | | funcrcl 17908 |
. . . . . 6
⊢ (𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸) → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat)) |
| 28 | 5, 27 | syl 17 |
. . . . 5
⊢ (𝜑 → ((𝐶 ×c 𝐷) ∈ Cat ∧ 𝐸 ∈ Cat)) |
| 29 | 28 | simprd 495 |
. . . 4
⊢ (𝜑 → 𝐸 ∈ Cat) |
| 30 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝐶 ×c
𝐷) = (𝐶 ×c 𝐷) |
| 31 | 30, 2, 6 | xpcbas 18223 |
. . . . . . . 8
⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷)) |
| 32 | | relfunc 17907 |
. . . . . . . . 9
⊢ Rel
((𝐶
×c 𝐷) Func 𝐸) |
| 33 | | 1st2ndbr 8067 |
. . . . . . . . 9
⊢ ((Rel
((𝐶
×c 𝐷) Func 𝐸) ∧ 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
| 34 | 32, 5, 33 | sylancr 587 |
. . . . . . . 8
⊢ (𝜑 → (1st
‘𝐹)((𝐶 ×c
𝐷) Func 𝐸)(2nd ‘𝐹)) |
| 35 | 31, 21, 34 | funcf1 17911 |
. . . . . . 7
⊢ (𝜑 → (1st
‘𝐹):(𝐴 × 𝐵)⟶(Base‘𝐸)) |
| 36 | 35 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (1st ‘𝐹):(𝐴 × 𝐵)⟶(Base‘𝐸)) |
| 37 | 7 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐴) |
| 38 | | simpr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) |
| 39 | 36, 37, 38 | fovcdmd 7605 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(1st ‘𝐹)𝑦) ∈ (Base‘𝐸)) |
| 40 | 16, 39 | fmpt3d 7136 |
. . . 4
⊢ (𝜑 → (1st
‘𝐾):𝐵⟶(Base‘𝐸)) |
| 41 | | eqid 2737 |
. . . . . 6
⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔))) |
| 42 | | ovex 7464 |
. . . . . . 7
⊢ (𝑦(Hom ‘𝐷)𝑧) ∈ V |
| 43 | 42 | mptex 7243 |
. . . . . 6
⊢ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)) ∈ V |
| 44 | 41, 43 | fnmpoi 8095 |
. . . . 5
⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔))) Fn (𝐵 × 𝐵) |
| 45 | 18 | fneq1d 6661 |
. . . . 5
⊢ (𝜑 → ((2nd
‘𝐾) Fn (𝐵 × 𝐵) ↔ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔))) Fn (𝐵 × 𝐵))) |
| 46 | 44, 45 | mpbiri 258 |
. . . 4
⊢ (𝜑 → (2nd
‘𝐾) Fn (𝐵 × 𝐵)) |
| 47 | 18 | oveqd 7448 |
. . . . . 6
⊢ (𝜑 → (𝑦(2nd ‘𝐾)𝑧) = (𝑦(𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)))𝑧)) |
| 48 | 41 | ovmpt4g 7580 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)) ∈ V) → (𝑦(𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)))𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔))) |
| 49 | 43, 48 | mp3an3 1452 |
. . . . . 6
⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)))𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔))) |
| 50 | 47, 49 | sylan9eq 2797 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(2nd ‘𝐾)𝑧) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔))) |
| 51 | | eqid 2737 |
. . . . . . . 8
⊢ (Hom
‘(𝐶
×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) |
| 52 | 34 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
| 53 | 7 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑋 ∈ 𝐴) |
| 54 | | simplrl 777 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦 ∈ 𝐵) |
| 55 | 53, 54 | opelxpd 5724 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 〈𝑋, 𝑦〉 ∈ (𝐴 × 𝐵)) |
| 56 | | simplrr 778 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧 ∈ 𝐵) |
| 57 | 53, 56 | opelxpd 5724 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 〈𝑋, 𝑧〉 ∈ (𝐴 × 𝐵)) |
| 58 | 31, 51, 22, 52, 55, 57 | funcf2 17913 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉):(〈𝑋, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑧〉)⟶(((1st ‘𝐹)‘〈𝑋, 𝑦〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑋, 𝑧〉))) |
| 59 | | eqid 2737 |
. . . . . . . . 9
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
| 60 | 30, 31, 59, 9, 51, 55, 57 | xpchom 18225 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (〈𝑋, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑧〉) = (((1st
‘〈𝑋, 𝑦〉)(Hom ‘𝐶)(1st
‘〈𝑋, 𝑧〉)) ×
((2nd ‘〈𝑋, 𝑦〉)(Hom ‘𝐷)(2nd ‘〈𝑋, 𝑧〉)))) |
| 61 | 3 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat) |
| 62 | 4 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat) |
| 63 | 5 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
| 64 | 1, 2, 61, 62, 63, 6, 53, 8, 54 | curf11 18271 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘𝐾)‘𝑦) = (𝑋(1st ‘𝐹)𝑦)) |
| 65 | | df-ov 7434 |
. . . . . . . . . 10
⊢ (𝑋(1st ‘𝐹)𝑦) = ((1st ‘𝐹)‘〈𝑋, 𝑦〉) |
| 66 | 64, 65 | eqtr2di 2794 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘𝐹)‘〈𝑋, 𝑦〉) = ((1st ‘𝐾)‘𝑦)) |
| 67 | 1, 2, 61, 62, 63, 6, 53, 8, 56 | curf11 18271 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘𝐾)‘𝑧) = (𝑋(1st ‘𝐹)𝑧)) |
| 68 | | df-ov 7434 |
. . . . . . . . . 10
⊢ (𝑋(1st ‘𝐹)𝑧) = ((1st ‘𝐹)‘〈𝑋, 𝑧〉) |
| 69 | 67, 68 | eqtr2di 2794 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘𝐹)‘〈𝑋, 𝑧〉) = ((1st ‘𝐾)‘𝑧)) |
| 70 | 66, 69 | oveq12d 7449 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((1st ‘𝐹)‘〈𝑋, 𝑦〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑋, 𝑧〉)) = (((1st ‘𝐾)‘𝑦)(Hom ‘𝐸)((1st ‘𝐾)‘𝑧))) |
| 71 | 60, 70 | feq23d 6731 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉):(〈𝑋, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑧〉)⟶(((1st ‘𝐹)‘〈𝑋, 𝑦〉)(Hom ‘𝐸)((1st ‘𝐹)‘〈𝑋, 𝑧〉)) ↔ (〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉):(((1st ‘〈𝑋, 𝑦〉)(Hom ‘𝐶)(1st ‘〈𝑋, 𝑧〉)) × ((2nd
‘〈𝑋, 𝑦〉)(Hom ‘𝐷)(2nd
‘〈𝑋, 𝑧〉)))⟶(((1st
‘𝐾)‘𝑦)(Hom ‘𝐸)((1st ‘𝐾)‘𝑧)))) |
| 72 | 58, 71 | mpbid 232 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉):(((1st ‘〈𝑋, 𝑦〉)(Hom ‘𝐶)(1st ‘〈𝑋, 𝑧〉)) × ((2nd
‘〈𝑋, 𝑦〉)(Hom ‘𝐷)(2nd
‘〈𝑋, 𝑧〉)))⟶(((1st
‘𝐾)‘𝑦)(Hom ‘𝐸)((1st ‘𝐾)‘𝑧))) |
| 73 | 2, 59, 10, 61, 53 | catidcl 17725 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
| 74 | | op1stg 8026 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (1st ‘〈𝑋, 𝑦〉) = 𝑋) |
| 75 | 53, 54, 74 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘〈𝑋, 𝑦〉) = 𝑋) |
| 76 | | op1stg 8026 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (1st ‘〈𝑋, 𝑧〉) = 𝑋) |
| 77 | 53, 56, 76 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (1st ‘〈𝑋, 𝑧〉) = 𝑋) |
| 78 | 75, 77 | oveq12d 7449 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((1st ‘〈𝑋, 𝑦〉)(Hom ‘𝐶)(1st ‘〈𝑋, 𝑧〉)) = (𝑋(Hom ‘𝐶)𝑋)) |
| 79 | 73, 78 | eleqtrrd 2844 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑋) ∈ ((1st ‘〈𝑋, 𝑦〉)(Hom ‘𝐶)(1st ‘〈𝑋, 𝑧〉))) |
| 80 | | simpr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) |
| 81 | | op2ndg 8027 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑦〉) = 𝑦) |
| 82 | 53, 54, 81 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘〈𝑋, 𝑦〉) = 𝑦) |
| 83 | | op2ndg 8027 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑧〉) = 𝑧) |
| 84 | 53, 56, 83 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (2nd ‘〈𝑋, 𝑧〉) = 𝑧) |
| 85 | 82, 84 | oveq12d 7449 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ‘〈𝑋, 𝑦〉)(Hom ‘𝐷)(2nd ‘〈𝑋, 𝑧〉)) = (𝑦(Hom ‘𝐷)𝑧)) |
| 86 | 80, 85 | eleqtrrd 2844 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑔 ∈ ((2nd ‘〈𝑋, 𝑦〉)(Hom ‘𝐷)(2nd ‘〈𝑋, 𝑧〉))) |
| 87 | 72, 79, 86 | fovcdmd 7605 |
. . . . 5
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔) ∈ (((1st ‘𝐾)‘𝑦)(Hom ‘𝐸)((1st ‘𝐾)‘𝑧))) |
| 88 | 50, 87 | fmpt3d 7136 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(2nd ‘𝐾)𝑧):(𝑦(Hom ‘𝐷)𝑧)⟶(((1st ‘𝐾)‘𝑦)(Hom ‘𝐸)((1st ‘𝐾)‘𝑧))) |
| 89 | 3 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat) |
| 90 | 4 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ Cat) |
| 91 | | eqid 2737 |
. . . . . . . . 9
⊢
(Id‘(𝐶
×c 𝐷)) = (Id‘(𝐶 ×c 𝐷)) |
| 92 | 30, 89, 90, 2, 6, 10, 23, 91, 37, 38 | xpcid 18234 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((Id‘(𝐶 ×c 𝐷))‘〈𝑋, 𝑦〉) = 〈((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)〉) |
| 93 | 92 | fveq2d 6910 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑦〉)‘((Id‘(𝐶 ×c 𝐷))‘〈𝑋, 𝑦〉)) = ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑦〉)‘〈((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)〉)) |
| 94 | | df-ov 7434 |
. . . . . . 7
⊢
(((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑦〉)((Id‘𝐷)‘𝑦)) = ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑦〉)‘〈((Id‘𝐶)‘𝑋), ((Id‘𝐷)‘𝑦)〉) |
| 95 | 93, 94 | eqtr4di 2795 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑦〉)‘((Id‘(𝐶 ×c 𝐷))‘〈𝑋, 𝑦〉)) = (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑦〉)((Id‘𝐷)‘𝑦))) |
| 96 | 34 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
| 97 | | opelxpi 5722 |
. . . . . . . 8
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 〈𝑋, 𝑦〉 ∈ (𝐴 × 𝐵)) |
| 98 | 7, 97 | sylan 580 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 〈𝑋, 𝑦〉 ∈ (𝐴 × 𝐵)) |
| 99 | 31, 91, 24, 96, 98 | funcid 17915 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑦〉)‘((Id‘(𝐶 ×c 𝐷))‘〈𝑋, 𝑦〉)) = ((Id‘𝐸)‘((1st ‘𝐹)‘〈𝑋, 𝑦〉))) |
| 100 | 95, 99 | eqtr3d 2779 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑦〉)((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st ‘𝐹)‘〈𝑋, 𝑦〉))) |
| 101 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
| 102 | 6, 9, 23, 90, 38 | catidcl 17725 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((Id‘𝐷)‘𝑦) ∈ (𝑦(Hom ‘𝐷)𝑦)) |
| 103 | 1, 2, 89, 90, 101, 6, 37, 8, 38, 9, 10, 38, 102 | curf12 18272 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑦(2nd ‘𝐾)𝑦)‘((Id‘𝐷)‘𝑦)) = (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑦〉)((Id‘𝐷)‘𝑦))) |
| 104 | 1, 2, 89, 90, 101, 6, 37, 8, 38 | curf11 18271 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = (𝑋(1st ‘𝐹)𝑦)) |
| 105 | 104, 65 | eqtrdi 2793 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = ((1st ‘𝐹)‘〈𝑋, 𝑦〉)) |
| 106 | 105 | fveq2d 6910 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((Id‘𝐸)‘((1st ‘𝐾)‘𝑦)) = ((Id‘𝐸)‘((1st ‘𝐹)‘〈𝑋, 𝑦〉))) |
| 107 | 100, 103,
106 | 3eqtr4d 2787 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑦(2nd ‘𝐾)𝑦)‘((Id‘𝐷)‘𝑦)) = ((Id‘𝐸)‘((1st ‘𝐾)‘𝑦))) |
| 108 | 7 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑋 ∈ 𝐴) |
| 109 | | simp21 1207 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑦 ∈ 𝐵) |
| 110 | | simp22 1208 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑧 ∈ 𝐵) |
| 111 | | eqid 2737 |
. . . . . . . . . 10
⊢
(comp‘𝐶) =
(comp‘𝐶) |
| 112 | | eqid 2737 |
. . . . . . . . . 10
⊢
(comp‘(𝐶
×c 𝐷)) = (comp‘(𝐶 ×c 𝐷)) |
| 113 | | simp23 1209 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑤 ∈ 𝐵) |
| 114 | 3 | 3ad2ant1 1134 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐶 ∈ Cat) |
| 115 | 2, 59, 10, 114, 108 | catidcl 17725 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋)) |
| 116 | | simp3l 1202 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)) |
| 117 | | simp3r 1203 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)) |
| 118 | 30, 2, 6, 59, 9, 108, 109, 108, 110, 111, 25, 112, 108, 113, 115, 116, 115, 117 | xpcco2 18232 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈((Id‘𝐶)‘𝑋), ℎ〉(〈〈𝑋, 𝑦〉, 〈𝑋, 𝑧〉〉(comp‘(𝐶 ×c 𝐷))〈𝑋, 𝑤〉)〈((Id‘𝐶)‘𝑋), 𝑔〉) = 〈(((Id‘𝐶)‘𝑋)(〈𝑋, 𝑋〉(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑋)), (ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔)〉) |
| 119 | 2, 59, 10, 114, 108, 111, 108, 115 | catlid 17726 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑋〉(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑋)) = ((Id‘𝐶)‘𝑋)) |
| 120 | 119 | opeq1d 4879 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈(((Id‘𝐶)‘𝑋)(〈𝑋, 𝑋〉(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑋)), (ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔)〉 = 〈((Id‘𝐶)‘𝑋), (ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔)〉) |
| 121 | 118, 120 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈((Id‘𝐶)‘𝑋), ℎ〉(〈〈𝑋, 𝑦〉, 〈𝑋, 𝑧〉〉(comp‘(𝐶 ×c 𝐷))〈𝑋, 𝑤〉)〈((Id‘𝐶)‘𝑋), 𝑔〉) = 〈((Id‘𝐶)‘𝑋), (ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔)〉) |
| 122 | 121 | fveq2d 6910 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘(〈((Id‘𝐶)‘𝑋), ℎ〉(〈〈𝑋, 𝑦〉, 〈𝑋, 𝑧〉〉(comp‘(𝐶 ×c 𝐷))〈𝑋, 𝑤〉)〈((Id‘𝐶)‘𝑋), 𝑔〉)) = ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), (ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔)〉)) |
| 123 | | df-ov 7434 |
. . . . . . 7
⊢
(((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑤〉)(ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔)) = ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), (ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔)〉) |
| 124 | 122, 123 | eqtr4di 2795 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘(〈((Id‘𝐶)‘𝑋), ℎ〉(〈〈𝑋, 𝑦〉, 〈𝑋, 𝑧〉〉(comp‘(𝐶 ×c 𝐷))〈𝑋, 𝑤〉)〈((Id‘𝐶)‘𝑋), 𝑔〉)) = (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑤〉)(ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔))) |
| 125 | 34 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (1st ‘𝐹)((𝐶 ×c 𝐷) Func 𝐸)(2nd ‘𝐹)) |
| 126 | 108, 109 | opelxpd 5724 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈𝑋, 𝑦〉 ∈ (𝐴 × 𝐵)) |
| 127 | 108, 110 | opelxpd 5724 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈𝑋, 𝑧〉 ∈ (𝐴 × 𝐵)) |
| 128 | 108, 113 | opelxpd 5724 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈𝑋, 𝑤〉 ∈ (𝐴 × 𝐵)) |
| 129 | 115, 116 | opelxpd 5724 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈((Id‘𝐶)‘𝑋), 𝑔〉 ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑦(Hom ‘𝐷)𝑧))) |
| 130 | 30, 2, 6, 59, 9, 108, 109, 108, 110, 51 | xpchom2 18231 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈𝑋, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑧〉) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑦(Hom ‘𝐷)𝑧))) |
| 131 | 129, 130 | eleqtrrd 2844 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈((Id‘𝐶)‘𝑋), 𝑔〉 ∈ (〈𝑋, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑧〉)) |
| 132 | 115, 117 | opelxpd 5724 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈((Id‘𝐶)‘𝑋), ℎ〉 ∈ ((𝑋(Hom ‘𝐶)𝑋) × (𝑧(Hom ‘𝐷)𝑤))) |
| 133 | 30, 2, 6, 59, 9, 108, 110, 108, 113, 51 | xpchom2 18231 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈𝑋, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑤〉) = ((𝑋(Hom ‘𝐶)𝑋) × (𝑧(Hom ‘𝐷)𝑤))) |
| 134 | 132, 133 | eleqtrrd 2844 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈((Id‘𝐶)‘𝑋), ℎ〉 ∈ (〈𝑋, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑋, 𝑤〉)) |
| 135 | 31, 51, 112, 26, 125, 126, 127, 128, 131, 134 | funcco 17916 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘(〈((Id‘𝐶)‘𝑋), ℎ〉(〈〈𝑋, 𝑦〉, 〈𝑋, 𝑧〉〉(comp‘(𝐶 ×c 𝐷))〈𝑋, 𝑤〉)〈((Id‘𝐶)‘𝑋), 𝑔〉)) = (((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), ℎ〉)(〈((1st ‘𝐹)‘〈𝑋, 𝑦〉), ((1st ‘𝐹)‘〈𝑋, 𝑧〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑋, 𝑤〉))((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)‘〈((Id‘𝐶)‘𝑋), 𝑔〉))) |
| 136 | 124, 135 | eqtr3d 2779 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑤〉)(ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔)) = (((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), ℎ〉)(〈((1st ‘𝐹)‘〈𝑋, 𝑦〉), ((1st ‘𝐹)‘〈𝑋, 𝑧〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑋, 𝑤〉))((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)‘〈((Id‘𝐶)‘𝑋), 𝑔〉))) |
| 137 | 4 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐷 ∈ Cat) |
| 138 | 5 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
| 139 | 6, 9, 25, 137, 109, 110, 113, 116, 117 | catcocl 17728 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐷)𝑤)) |
| 140 | 1, 2, 114, 137, 138, 6, 108, 8, 109, 9, 10, 113, 139 | curf12 18272 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘𝐾)𝑤)‘(ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔)) = (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑤〉)(ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔))) |
| 141 | 1, 2, 114, 137, 138, 6, 108, 8, 109 | curf11 18271 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑦) = (𝑋(1st ‘𝐹)𝑦)) |
| 142 | 141, 65 | eqtrdi 2793 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑦) = ((1st ‘𝐹)‘〈𝑋, 𝑦〉)) |
| 143 | 1, 2, 114, 137, 138, 6, 108, 8, 110 | curf11 18271 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑧) = (𝑋(1st ‘𝐹)𝑧)) |
| 144 | 143, 68 | eqtrdi 2793 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑧) = ((1st ‘𝐹)‘〈𝑋, 𝑧〉)) |
| 145 | 142, 144 | opeq12d 4881 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → 〈((1st
‘𝐾)‘𝑦), ((1st ‘𝐾)‘𝑧)〉 = 〈((1st ‘𝐹)‘〈𝑋, 𝑦〉), ((1st ‘𝐹)‘〈𝑋, 𝑧〉)〉) |
| 146 | 1, 2, 114, 137, 138, 6, 108, 8, 113 | curf11 18271 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑤) = (𝑋(1st ‘𝐹)𝑤)) |
| 147 | | df-ov 7434 |
. . . . . . . 8
⊢ (𝑋(1st ‘𝐹)𝑤) = ((1st ‘𝐹)‘〈𝑋, 𝑤〉) |
| 148 | 146, 147 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((1st ‘𝐾)‘𝑤) = ((1st ‘𝐹)‘〈𝑋, 𝑤〉)) |
| 149 | 145, 148 | oveq12d 7449 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (〈((1st
‘𝐾)‘𝑦), ((1st ‘𝐾)‘𝑧)〉(comp‘𝐸)((1st ‘𝐾)‘𝑤)) = (〈((1st ‘𝐹)‘〈𝑋, 𝑦〉), ((1st ‘𝐹)‘〈𝑋, 𝑧〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑋, 𝑤〉))) |
| 150 | 1, 2, 114, 137, 138, 6, 108, 8, 110, 9, 10, 113, 117 | curf12 18272 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘𝐾)𝑤)‘ℎ) = (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)ℎ)) |
| 151 | | df-ov 7434 |
. . . . . . 7
⊢
(((Id‘𝐶)‘𝑋)(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)ℎ) = ((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), ℎ〉) |
| 152 | 150, 151 | eqtrdi 2793 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑧(2nd ‘𝐾)𝑤)‘ℎ) = ((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), ℎ〉)) |
| 153 | 1, 2, 114, 137, 138, 6, 108, 8, 109, 9, 10, 110, 116 | curf12 18272 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘𝐾)𝑧)‘𝑔) = (((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)) |
| 154 | | df-ov 7434 |
. . . . . . 7
⊢
(((Id‘𝐶)‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔) = ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)‘〈((Id‘𝐶)‘𝑋), 𝑔〉) |
| 155 | 153, 154 | eqtrdi 2793 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘𝐾)𝑧)‘𝑔) = ((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)‘〈((Id‘𝐶)‘𝑋), 𝑔〉)) |
| 156 | 149, 152,
155 | oveq123d 7452 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → (((𝑧(2nd ‘𝐾)𝑤)‘ℎ)(〈((1st ‘𝐾)‘𝑦), ((1st ‘𝐾)‘𝑧)〉(comp‘𝐸)((1st ‘𝐾)‘𝑤))((𝑦(2nd ‘𝐾)𝑧)‘𝑔)) = (((〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑋, 𝑤〉)‘〈((Id‘𝐶)‘𝑋), ℎ〉)(〈((1st ‘𝐹)‘〈𝑋, 𝑦〉), ((1st ‘𝐹)‘〈𝑋, 𝑧〉)〉(comp‘𝐸)((1st ‘𝐹)‘〈𝑋, 𝑤〉))((〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)‘〈((Id‘𝐶)‘𝑋), 𝑔〉))) |
| 157 | 136, 140,
156 | 3eqtr4d 2787 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐷)𝑤))) → ((𝑦(2nd ‘𝐾)𝑤)‘(ℎ(〈𝑦, 𝑧〉(comp‘𝐷)𝑤)𝑔)) = (((𝑧(2nd ‘𝐾)𝑤)‘ℎ)(〈((1st ‘𝐾)‘𝑦), ((1st ‘𝐾)‘𝑧)〉(comp‘𝐸)((1st ‘𝐾)‘𝑤))((𝑦(2nd ‘𝐾)𝑧)‘𝑔))) |
| 158 | 6, 21, 9, 22, 23, 24, 25, 26, 4, 29, 40, 46, 88, 107, 157 | isfuncd 17910 |
. . 3
⊢ (𝜑 → (1st
‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾)) |
| 159 | | df-br 5144 |
. . 3
⊢
((1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾) ↔ 〈(1st ‘𝐾), (2nd ‘𝐾)〉 ∈ (𝐷 Func 𝐸)) |
| 160 | 158, 159 | sylib 218 |
. 2
⊢ (𝜑 → 〈(1st
‘𝐾), (2nd
‘𝐾)〉 ∈
(𝐷 Func 𝐸)) |
| 161 | 20, 160 | eqeltrd 2841 |
1
⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸)) |