| Step | Hyp | Ref
| Expression |
| 1 | | funcres.f |
. . . 4
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
| 2 | | funcres.h |
. . . 4
⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) |
| 3 | 1, 2 | resfval 17937 |
. . 3
⊢ (𝜑 → (𝐹 ↾f 𝐻) = 〈((1st
‘𝐹) ↾ dom dom
𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉) |
| 4 | 3 | fveq2d 6910 |
. . . . 5
⊢ (𝜑 → (2nd
‘(𝐹
↾f 𝐻)) = (2nd
‘〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉)) |
| 5 | | fvex 6919 |
. . . . . . 7
⊢
(1st ‘𝐹) ∈ V |
| 6 | 5 | resex 6047 |
. . . . . 6
⊢
((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V |
| 7 | | dmexg 7923 |
. . . . . . 7
⊢ (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V) |
| 8 | | mptexg 7241 |
. . . . . . 7
⊢ (dom
𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V) |
| 9 | 2, 7, 8 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V) |
| 10 | | op2ndg 8027 |
. . . . . 6
⊢
((((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V ∧ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V) → (2nd
‘〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))) |
| 11 | 6, 9, 10 | sylancr 587 |
. . . . 5
⊢ (𝜑 → (2nd
‘〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))) |
| 12 | 4, 11 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → (2nd
‘(𝐹
↾f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))) |
| 13 | 12 | opeq2d 4880 |
. . 3
⊢ (𝜑 → 〈((1st
‘𝐹) ↾ dom dom
𝐻), (2nd
‘(𝐹
↾f 𝐻))〉 = 〈((1st
‘𝐹) ↾ dom dom
𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉) |
| 14 | 3, 13 | eqtr4d 2780 |
. 2
⊢ (𝜑 → (𝐹 ↾f 𝐻) = 〈((1st
‘𝐹) ↾ dom dom
𝐻), (2nd
‘(𝐹
↾f 𝐻))〉) |
| 15 | | eqid 2737 |
. . . 4
⊢
(Base‘(𝐶
↾cat 𝐻)) =
(Base‘(𝐶
↾cat 𝐻)) |
| 16 | | eqid 2737 |
. . . 4
⊢
(Base‘𝐷) =
(Base‘𝐷) |
| 17 | | eqid 2737 |
. . . 4
⊢ (Hom
‘(𝐶
↾cat 𝐻)) =
(Hom ‘(𝐶
↾cat 𝐻)) |
| 18 | | eqid 2737 |
. . . 4
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) |
| 19 | | eqid 2737 |
. . . 4
⊢
(Id‘(𝐶
↾cat 𝐻)) =
(Id‘(𝐶
↾cat 𝐻)) |
| 20 | | eqid 2737 |
. . . 4
⊢
(Id‘𝐷) =
(Id‘𝐷) |
| 21 | | eqid 2737 |
. . . 4
⊢
(comp‘(𝐶
↾cat 𝐻)) =
(comp‘(𝐶
↾cat 𝐻)) |
| 22 | | eqid 2737 |
. . . 4
⊢
(comp‘𝐷) =
(comp‘𝐷) |
| 23 | | eqid 2737 |
. . . . 5
⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) |
| 24 | 23, 2 | subccat 17893 |
. . . 4
⊢ (𝜑 → (𝐶 ↾cat 𝐻) ∈ Cat) |
| 25 | | funcrcl 17908 |
. . . . . 6
⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) |
| 26 | 1, 25 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) |
| 27 | 26 | simprd 495 |
. . . 4
⊢ (𝜑 → 𝐷 ∈ Cat) |
| 28 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝐶) =
(Base‘𝐶) |
| 29 | | relfunc 17907 |
. . . . . . . 8
⊢ Rel
(𝐶 Func 𝐷) |
| 30 | | 1st2ndbr 8067 |
. . . . . . . 8
⊢ ((Rel
(𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
| 31 | 29, 1, 30 | sylancr 587 |
. . . . . . 7
⊢ (𝜑 → (1st
‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
| 32 | 28, 16, 31 | funcf1 17911 |
. . . . . 6
⊢ (𝜑 → (1st
‘𝐹):(Base‘𝐶)⟶(Base‘𝐷)) |
| 33 | | eqidd 2738 |
. . . . . . . 8
⊢ (𝜑 → dom dom 𝐻 = dom dom 𝐻) |
| 34 | 2, 33 | subcfn 17886 |
. . . . . . 7
⊢ (𝜑 → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻)) |
| 35 | 2, 34, 28 | subcss1 17887 |
. . . . . 6
⊢ (𝜑 → dom dom 𝐻 ⊆ (Base‘𝐶)) |
| 36 | 32, 35 | fssresd 6775 |
. . . . 5
⊢ (𝜑 → ((1st
‘𝐹) ↾ dom dom
𝐻):dom dom 𝐻⟶(Base‘𝐷)) |
| 37 | 26 | simpld 494 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ Cat) |
| 38 | 23, 28, 37, 34, 35 | rescbas 17873 |
. . . . . 6
⊢ (𝜑 → dom dom 𝐻 = (Base‘(𝐶 ↾cat 𝐻))) |
| 39 | 38 | feq2d 6722 |
. . . . 5
⊢ (𝜑 → (((1st
‘𝐹) ↾ dom dom
𝐻):dom dom 𝐻⟶(Base‘𝐷) ↔ ((1st
‘𝐹) ↾ dom dom
𝐻):(Base‘(𝐶 ↾cat 𝐻))⟶(Base‘𝐷))) |
| 40 | 36, 39 | mpbid 232 |
. . . 4
⊢ (𝜑 → ((1st
‘𝐹) ↾ dom dom
𝐻):(Base‘(𝐶 ↾cat 𝐻))⟶(Base‘𝐷)) |
| 41 | | fvex 6919 |
. . . . . . 7
⊢
((2nd ‘𝐹)‘𝑧) ∈ V |
| 42 | 41 | resex 6047 |
. . . . . 6
⊢
(((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)) ∈ V |
| 43 | | eqid 2737 |
. . . . . 6
⊢ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) |
| 44 | 42, 43 | fnmpti 6711 |
. . . . 5
⊢ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) Fn dom 𝐻 |
| 45 | 12 | eqcomd 2743 |
. . . . . 6
⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) = (2nd ‘(𝐹 ↾f
𝐻))) |
| 46 | | fndm 6671 |
. . . . . . . 8
⊢ (𝐻 Fn (dom dom 𝐻 × dom dom 𝐻) → dom 𝐻 = (dom dom 𝐻 × dom dom 𝐻)) |
| 47 | 34, 46 | syl 17 |
. . . . . . 7
⊢ (𝜑 → dom 𝐻 = (dom dom 𝐻 × dom dom 𝐻)) |
| 48 | 38 | sqxpeqd 5717 |
. . . . . . 7
⊢ (𝜑 → (dom dom 𝐻 × dom dom 𝐻) = ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻)))) |
| 49 | 47, 48 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → dom 𝐻 = ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻)))) |
| 50 | 45, 49 | fneq12d 6663 |
. . . . 5
⊢ (𝜑 → ((𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) Fn dom 𝐻 ↔ (2nd ‘(𝐹 ↾f
𝐻)) Fn ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻))))) |
| 51 | 44, 50 | mpbii 233 |
. . . 4
⊢ (𝜑 → (2nd
‘(𝐹
↾f 𝐻)) Fn ((Base‘(𝐶 ↾cat 𝐻)) × (Base‘(𝐶 ↾cat 𝐻)))) |
| 52 | | eqid 2737 |
. . . . . . . 8
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
| 53 | 31 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
| 54 | 35 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → dom dom 𝐻 ⊆ (Base‘𝐶)) |
| 55 | | simprl 771 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) |
| 56 | 38 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → dom dom 𝐻 = (Base‘(𝐶 ↾cat 𝐻))) |
| 57 | 55, 56 | eleqtrrd 2844 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑥 ∈ dom dom 𝐻) |
| 58 | 54, 57 | sseldd 3984 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑥 ∈ (Base‘𝐶)) |
| 59 | | simprr 773 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻))) |
| 60 | 59, 56 | eleqtrrd 2844 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑦 ∈ dom dom 𝐻) |
| 61 | 54, 60 | sseldd 3984 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝑦 ∈ (Base‘𝐶)) |
| 62 | 28, 52, 18, 53, 58, 61 | funcf2 17913 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))) |
| 63 | 2 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝐻 ∈ (Subcat‘𝐶)) |
| 64 | 34 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻)) |
| 65 | 63, 64, 52, 57, 60 | subcss2 17888 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥𝐻𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦)) |
| 66 | 62, 65 | fssresd 6775 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → ((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦)):(𝑥𝐻𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))) |
| 67 | 1 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝐹 ∈ (𝐶 Func 𝐷)) |
| 68 | 67, 63, 64, 57, 60 | resf2nd 17940 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦) = ((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦))) |
| 69 | 68 | feq1d 6720 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦):(𝑥𝐻𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)) ↔ ((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦)):(𝑥𝐻𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))) |
| 70 | 66, 69 | mpbird 257 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦):(𝑥𝐻𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))) |
| 71 | 23, 28, 37, 34, 35 | reschom 17874 |
. . . . . . . 8
⊢ (𝜑 → 𝐻 = (Hom ‘(𝐶 ↾cat 𝐻))) |
| 72 | 71 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → 𝐻 = (Hom ‘(𝐶 ↾cat 𝐻))) |
| 73 | 72 | oveqd 7448 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥𝐻𝑦) = (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦)) |
| 74 | 57 | fvresd 6926 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st ‘𝐹)‘𝑥)) |
| 75 | 60 | fvresd 6926 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦) = ((1st ‘𝐹)‘𝑦)) |
| 76 | 74, 75 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → ((((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)) = (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))) |
| 77 | 76 | eqcomd 2743 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)) = ((((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦))) |
| 78 | 73, 77 | feq23d 6731 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦):(𝑥𝐻𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)) ↔ (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦):(𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦)⟶((((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)))) |
| 79 | 70, 78 | mpbid 232 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦):(𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦)⟶((((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)(Hom ‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦))) |
| 80 | 1 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝐹 ∈ (𝐶 Func 𝐷)) |
| 81 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝐻 ∈ (Subcat‘𝐶)) |
| 82 | 34 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻)) |
| 83 | 38 | eleq2d 2827 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ dom dom 𝐻 ↔ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)))) |
| 84 | 83 | biimpar 477 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝑥 ∈ dom dom 𝐻) |
| 85 | 80, 81, 82, 84, 84 | resf2nd 17940 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑥) = ((𝑥(2nd ‘𝐹)𝑥) ↾ (𝑥𝐻𝑥))) |
| 86 | | eqid 2737 |
. . . . . . . 8
⊢
(Id‘𝐶) =
(Id‘𝐶) |
| 87 | 23, 81, 82, 86, 84 | subcid 17892 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘𝐶)‘𝑥) = ((Id‘(𝐶 ↾cat 𝐻))‘𝑥)) |
| 88 | 87 | eqcomd 2743 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘(𝐶 ↾cat 𝐻))‘𝑥) = ((Id‘𝐶)‘𝑥)) |
| 89 | 85, 88 | fveq12d 6913 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑥)‘((Id‘(𝐶 ↾cat 𝐻))‘𝑥)) = (((𝑥(2nd ‘𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥))) |
| 90 | 31 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
| 91 | 38, 35 | eqsstrrd 4019 |
. . . . . . . 8
⊢ (𝜑 → (Base‘(𝐶 ↾cat 𝐻)) ⊆ (Base‘𝐶)) |
| 92 | 91 | sselda 3983 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → 𝑥 ∈ (Base‘𝐶)) |
| 93 | 28, 86, 20, 90, 92 | funcid 17915 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((𝑥(2nd ‘𝐹)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘((1st ‘𝐹)‘𝑥))) |
| 94 | 81, 82, 84, 86 | subcidcl 17889 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
| 95 | 94 | fvresd 6926 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (((𝑥(2nd ‘𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)) = ((𝑥(2nd ‘𝐹)𝑥)‘((Id‘𝐶)‘𝑥))) |
| 96 | 84 | fvresd 6926 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st ‘𝐹)‘𝑥)) |
| 97 | 96 | fveq2d 6910 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((Id‘𝐷)‘(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥)) = ((Id‘𝐷)‘((1st ‘𝐹)‘𝑥))) |
| 98 | 93, 95, 97 | 3eqtr4d 2787 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → (((𝑥(2nd ‘𝐹)𝑥) ↾ (𝑥𝐻𝑥))‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥))) |
| 99 | 89, 98 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑥)‘((Id‘(𝐶 ↾cat 𝐻))‘𝑥)) = ((Id‘𝐷)‘(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥))) |
| 100 | 2 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝐻 ∈ (Subcat‘𝐶)) |
| 101 | 34 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻)) |
| 102 | | simp21 1207 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻))) |
| 103 | 38 | 3ad2ant1 1134 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → dom dom 𝐻 = (Base‘(𝐶 ↾cat 𝐻))) |
| 104 | 102, 103 | eleqtrrd 2844 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑥 ∈ dom dom 𝐻) |
| 105 | | eqid 2737 |
. . . . . . . 8
⊢
(comp‘𝐶) =
(comp‘𝐶) |
| 106 | | simp22 1208 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻))) |
| 107 | 106, 103 | eleqtrrd 2844 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑦 ∈ dom dom 𝐻) |
| 108 | | simp23 1209 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) |
| 109 | 108, 103 | eleqtrrd 2844 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑧 ∈ dom dom 𝐻) |
| 110 | | simp3l 1202 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦)) |
| 111 | 71 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝐻 = (Hom ‘(𝐶 ↾cat 𝐻))) |
| 112 | 111 | oveqd 7448 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑥𝐻𝑦) = (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦)) |
| 113 | 110, 112 | eleqtrrd 2844 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑓 ∈ (𝑥𝐻𝑦)) |
| 114 | | simp3r 1203 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧)) |
| 115 | 111 | oveqd 7448 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑦𝐻𝑧) = (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧)) |
| 116 | 114, 115 | eleqtrrd 2844 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑔 ∈ (𝑦𝐻𝑧)) |
| 117 | 100, 101,
104, 105, 107, 109, 113, 116 | subccocl 17890 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) |
| 118 | 117 | fvresd 6926 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((𝑥(2nd ‘𝐹)𝑧) ↾ (𝑥𝐻𝑧))‘(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓)) = ((𝑥(2nd ‘𝐹)𝑧)‘(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓))) |
| 119 | 31 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
| 120 | 35 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → dom dom 𝐻 ⊆ (Base‘𝐶)) |
| 121 | 120, 104 | sseldd 3984 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑥 ∈ (Base‘𝐶)) |
| 122 | 120, 107 | sseldd 3984 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑦 ∈ (Base‘𝐶)) |
| 123 | 120, 109 | sseldd 3984 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑧 ∈ (Base‘𝐶)) |
| 124 | 100, 101,
52, 104, 107 | subcss2 17888 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑥𝐻𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦)) |
| 125 | 124, 113 | sseldd 3984 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) |
| 126 | 100, 101,
52, 107, 109 | subcss2 17888 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑦𝐻𝑧) ⊆ (𝑦(Hom ‘𝐶)𝑧)) |
| 127 | 126, 116 | sseldd 3984 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) |
| 128 | 28, 52, 105, 22, 119, 121, 122, 123, 125, 127 | funcco 17916 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑥(2nd ‘𝐹)𝑧)‘(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd ‘𝐹)𝑧)‘𝑔)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)〉(comp‘𝐷)((1st ‘𝐹)‘𝑧))((𝑥(2nd ‘𝐹)𝑦)‘𝑓))) |
| 129 | 118, 128 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((𝑥(2nd ‘𝐹)𝑧) ↾ (𝑥𝐻𝑧))‘(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd ‘𝐹)𝑧)‘𝑔)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)〉(comp‘𝐷)((1st ‘𝐹)‘𝑧))((𝑥(2nd ‘𝐹)𝑦)‘𝑓))) |
| 130 | 1 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 𝐹 ∈ (𝐶 Func 𝐷)) |
| 131 | 130, 100,
101, 104, 109 | resf2nd 17940 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑧) = ((𝑥(2nd ‘𝐹)𝑧) ↾ (𝑥𝐻𝑧))) |
| 132 | 23, 28, 37, 34, 35, 105 | rescco 17876 |
. . . . . . . . . 10
⊢ (𝜑 → (comp‘𝐶) = (comp‘(𝐶 ↾cat 𝐻))) |
| 133 | 132 | 3ad2ant1 1134 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (comp‘𝐶) = (comp‘(𝐶 ↾cat 𝐻))) |
| 134 | 133 | eqcomd 2743 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (comp‘(𝐶 ↾cat 𝐻)) = (comp‘𝐶)) |
| 135 | 134 | oveqd 7448 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (〈𝑥, 𝑦〉(comp‘(𝐶 ↾cat 𝐻))𝑧) = (〈𝑥, 𝑦〉(comp‘𝐶)𝑧)) |
| 136 | 135 | oveqd 7448 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑔(〈𝑥, 𝑦〉(comp‘(𝐶 ↾cat 𝐻))𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓)) |
| 137 | 131, 136 | fveq12d 6913 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘(𝑔(〈𝑥, 𝑦〉(comp‘(𝐶 ↾cat 𝐻))𝑧)𝑓)) = (((𝑥(2nd ‘𝐹)𝑧) ↾ (𝑥𝐻𝑧))‘(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓))) |
| 138 | 104 | fvresd 6926 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥) = ((1st ‘𝐹)‘𝑥)) |
| 139 | 107 | fvresd 6926 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦) = ((1st ‘𝐹)‘𝑦)) |
| 140 | 138, 139 | opeq12d 4881 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → 〈(((1st
‘𝐹) ↾ dom dom
𝐻)‘𝑥), (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)〉 = 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)〉) |
| 141 | 109 | fvresd 6926 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑧) = ((1st ‘𝐹)‘𝑧)) |
| 142 | 140, 141 | oveq12d 7449 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (〈(((1st
‘𝐹) ↾ dom dom
𝐻)‘𝑥), (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)〉(comp‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑧)) = (〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)〉(comp‘𝐷)((1st ‘𝐹)‘𝑧))) |
| 143 | 130, 100,
101, 107, 109 | resf2nd 17940 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑦(2nd ‘(𝐹 ↾f 𝐻))𝑧) = ((𝑦(2nd ‘𝐹)𝑧) ↾ (𝑦𝐻𝑧))) |
| 144 | 143 | fveq1d 6908 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑦(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘𝑔) = (((𝑦(2nd ‘𝐹)𝑧) ↾ (𝑦𝐻𝑧))‘𝑔)) |
| 145 | 116 | fvresd 6926 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((𝑦(2nd ‘𝐹)𝑧) ↾ (𝑦𝐻𝑧))‘𝑔) = ((𝑦(2nd ‘𝐹)𝑧)‘𝑔)) |
| 146 | 144, 145 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑦(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘𝑔) = ((𝑦(2nd ‘𝐹)𝑧)‘𝑔)) |
| 147 | 130, 100,
101, 104, 107 | resf2nd 17940 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦) = ((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦))) |
| 148 | 147 | fveq1d 6908 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦)‘𝑓) = (((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦))‘𝑓)) |
| 149 | 113 | fvresd 6926 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((𝑥(2nd ‘𝐹)𝑦) ↾ (𝑥𝐻𝑦))‘𝑓) = ((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) |
| 150 | 148, 149 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦)‘𝑓) = ((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) |
| 151 | 142, 146,
150 | oveq123d 7452 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → (((𝑦(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘𝑔)(〈(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥), (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)〉(comp‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑧))((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦)‘𝑓)) = (((𝑦(2nd ‘𝐹)𝑧)‘𝑔)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)〉(comp‘𝐷)((1st ‘𝐹)‘𝑧))((𝑥(2nd ‘𝐹)𝑦)‘𝑓))) |
| 152 | 129, 137,
151 | 3eqtr4d 2787 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑦 ∈ (Base‘(𝐶 ↾cat 𝐻)) ∧ 𝑧 ∈ (Base‘(𝐶 ↾cat 𝐻))) ∧ (𝑓 ∈ (𝑥(Hom ‘(𝐶 ↾cat 𝐻))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(𝐶 ↾cat 𝐻))𝑧))) → ((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘(𝑔(〈𝑥, 𝑦〉(comp‘(𝐶 ↾cat 𝐻))𝑧)𝑓)) = (((𝑦(2nd ‘(𝐹 ↾f 𝐻))𝑧)‘𝑔)(〈(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑥), (((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑦)〉(comp‘𝐷)(((1st ‘𝐹) ↾ dom dom 𝐻)‘𝑧))((𝑥(2nd ‘(𝐹 ↾f 𝐻))𝑦)‘𝑓))) |
| 153 | 15, 16, 17, 18, 19, 20, 21, 22, 24, 27, 40, 51, 79, 99, 152 | isfuncd 17910 |
. . 3
⊢ (𝜑 → ((1st
‘𝐹) ↾ dom dom
𝐻)((𝐶 ↾cat 𝐻) Func 𝐷)(2nd ‘(𝐹 ↾f 𝐻))) |
| 154 | | df-br 5144 |
. . 3
⊢
(((1st ‘𝐹) ↾ dom dom 𝐻)((𝐶 ↾cat 𝐻) Func 𝐷)(2nd ‘(𝐹 ↾f 𝐻)) ↔ 〈((1st
‘𝐹) ↾ dom dom
𝐻), (2nd
‘(𝐹
↾f 𝐻))〉 ∈ ((𝐶 ↾cat 𝐻) Func 𝐷)) |
| 155 | 153, 154 | sylib 218 |
. 2
⊢ (𝜑 → 〈((1st
‘𝐹) ↾ dom dom
𝐻), (2nd
‘(𝐹
↾f 𝐻))〉 ∈ ((𝐶 ↾cat 𝐻) Func 𝐷)) |
| 156 | 14, 155 | eqeltrd 2841 |
1
⊢ (𝜑 → (𝐹 ↾f 𝐻) ∈ ((𝐶 ↾cat 𝐻) Func 𝐷)) |