| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-ov 7434 | . . . . . . . . . . 11
⊢ (𝑥(1st ‘(𝐶
2ndF 𝐷))𝑦) = ((1st ‘(𝐶
2ndF 𝐷))‘〈𝑥, 𝑦〉) | 
| 2 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢ (𝐶 ×c
𝐷) = (𝐶 ×c 𝐷) | 
| 3 |  | eqid 2737 | . . . . . . . . . . . . . 14
⊢
(Base‘𝐶) =
(Base‘𝐶) | 
| 4 |  | eqid 2737 | . . . . . . . . . . . . . 14
⊢
(Base‘𝐷) =
(Base‘𝐷) | 
| 5 | 2, 3, 4 | xpcbas 18223 | . . . . . . . . . . . . 13
⊢
((Base‘𝐶)
× (Base‘𝐷)) =
(Base‘(𝐶
×c 𝐷)) | 
| 6 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢ (Hom
‘(𝐶
×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) | 
| 7 |  | curf2ndf.c | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐶 ∈ Cat) | 
| 8 | 7 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat) | 
| 9 |  | curf2ndf.d | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐷 ∈ Cat) | 
| 10 | 9 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat) | 
| 11 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢ (𝐶
2ndF 𝐷) = (𝐶 2ndF 𝐷) | 
| 12 |  | opelxpi 5722 | . . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐷)) → 〈𝑥, 𝑦〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) | 
| 13 | 12 | adantll 714 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → 〈𝑥, 𝑦〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) | 
| 14 | 2, 5, 6, 8, 10, 11, 13 | 2ndf1 18240 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘(𝐶
2ndF 𝐷))‘〈𝑥, 𝑦〉) = (2nd ‘〈𝑥, 𝑦〉)) | 
| 15 |  | vex 3484 | . . . . . . . . . . . . 13
⊢ 𝑥 ∈ V | 
| 16 |  | vex 3484 | . . . . . . . . . . . . 13
⊢ 𝑦 ∈ V | 
| 17 | 15, 16 | op2nd 8023 | . . . . . . . . . . . 12
⊢
(2nd ‘〈𝑥, 𝑦〉) = 𝑦 | 
| 18 | 14, 17 | eqtrdi 2793 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘(𝐶
2ndF 𝐷))‘〈𝑥, 𝑦〉) = 𝑦) | 
| 19 | 1, 18 | eqtrid 2789 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) → (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦) = 𝑦) | 
| 20 | 19 | mpteq2dva 5242 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)) = (𝑦 ∈ (Base‘𝐷) ↦ 𝑦)) | 
| 21 |  | mptresid 6069 | . . . . . . . . 9
⊢ ( I
↾ (Base‘𝐷)) =
(𝑦 ∈ (Base‘𝐷) ↦ 𝑦) | 
| 22 | 20, 21 | eqtr4di 2795 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)) = ( I ↾ (Base‘𝐷))) | 
| 23 |  | df-ov 7434 | . . . . . . . . . . . . . . 15
⊢
(((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓) = ((〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)‘〈((Id‘𝐶)‘𝑥), 𝑓〉) | 
| 24 | 8 | ad2antrr 726 | . . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐶 ∈ Cat) | 
| 25 | 10 | ad2antrr 726 | . . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝐷 ∈ Cat) | 
| 26 | 13 | ad2antrr 726 | . . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 〈𝑥, 𝑦〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) | 
| 27 |  | simp-4r 784 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑥 ∈ (Base‘𝐶)) | 
| 28 |  | simplr 769 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑧 ∈ (Base‘𝐷)) | 
| 29 | 27, 28 | opelxpd 5724 | . . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 〈𝑥, 𝑧〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) | 
| 30 | 2, 5, 6, 24, 25, 11, 26, 29 | 2ndf2 18241 | . . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉) = (2nd ↾
(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c
𝐷))〈𝑥, 𝑧〉))) | 
| 31 | 30 | fveq1d 6908 | . . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)‘〈((Id‘𝐶)‘𝑥), 𝑓〉) = ((2nd ↾
(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c
𝐷))〈𝑥, 𝑧〉))‘〈((Id‘𝐶)‘𝑥), 𝑓〉)) | 
| 32 | 23, 31 | eqtrid 2789 | . . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓) = ((2nd ↾ (〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑥, 𝑧〉))‘〈((Id‘𝐶)‘𝑥), 𝑓〉)) | 
| 33 |  | eqid 2737 | . . . . . . . . . . . . . . . . . . . 20
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) | 
| 34 |  | eqid 2737 | . . . . . . . . . . . . . . . . . . . 20
⊢
(Id‘𝐶) =
(Id‘𝐶) | 
| 35 | 7 | adantr 480 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat) | 
| 36 |  | simpr 484 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶)) | 
| 37 | 3, 33, 34, 35, 36 | catidcl 17725 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥)) | 
| 38 | 37 | ad5ant12 756 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥)) | 
| 39 |  | simpr 484 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) | 
| 40 | 38, 39 | opelxpd 5724 | . . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 〈((Id‘𝐶)‘𝑥), 𝑓〉 ∈ ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑧))) | 
| 41 |  | eqid 2737 | . . . . . . . . . . . . . . . . . 18
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) | 
| 42 |  | simpllr 776 | . . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 𝑦 ∈ (Base‘𝐷)) | 
| 43 | 2, 3, 4, 33, 41, 27, 42, 27, 28, 6 | xpchom2 18231 | . . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑥, 𝑧〉) = ((𝑥(Hom ‘𝐶)𝑥) × (𝑦(Hom ‘𝐷)𝑧))) | 
| 44 | 40, 43 | eleqtrrd 2844 | . . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → 〈((Id‘𝐶)‘𝑥), 𝑓〉 ∈ (〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c 𝐷))〈𝑥, 𝑧〉)) | 
| 45 | 44 | fvresd 6926 | . . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ↾
(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c
𝐷))〈𝑥, 𝑧〉))‘〈((Id‘𝐶)‘𝑥), 𝑓〉) = (2nd
‘〈((Id‘𝐶)‘𝑥), 𝑓〉)) | 
| 46 |  | fvex 6919 | . . . . . . . . . . . . . . . 16
⊢
((Id‘𝐶)‘𝑥) ∈ V | 
| 47 |  | vex 3484 | . . . . . . . . . . . . . . . 16
⊢ 𝑓 ∈ V | 
| 48 | 46, 47 | op2nd 8023 | . . . . . . . . . . . . . . 15
⊢
(2nd ‘〈((Id‘𝐶)‘𝑥), 𝑓〉) = 𝑓 | 
| 49 | 45, 48 | eqtrdi 2793 | . . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → ((2nd ↾
(〈𝑥, 𝑦〉(Hom ‘(𝐶 ×c
𝐷))〈𝑥, 𝑧〉))‘〈((Id‘𝐶)‘𝑥), 𝑓〉) = 𝑓) | 
| 50 | 32, 49 | eqtrd 2777 | . . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧)) → (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓) = 𝑓) | 
| 51 | 50 | mpteq2dva 5242 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓)) = (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ 𝑓)) | 
| 52 |  | mptresid 6069 | . . . . . . . . . . . 12
⊢ ( I
↾ (𝑦(Hom ‘𝐷)𝑧)) = (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ 𝑓) | 
| 53 | 51, 52 | eqtr4di 2795 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓)) = ( I ↾ (𝑦(Hom ‘𝐷)𝑧))) | 
| 54 | 53 | 3impa 1110 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓)) = ( I ↾ (𝑦(Hom ‘𝐷)𝑧))) | 
| 55 | 54 | mpoeq3dva 7510 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓))) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ ( I ↾ (𝑦(Hom ‘𝐷)𝑧)))) | 
| 56 |  | fveq2 6906 | . . . . . . . . . . . 12
⊢ (𝑢 = 〈𝑦, 𝑧〉 → ((Hom ‘𝐷)‘𝑢) = ((Hom ‘𝐷)‘〈𝑦, 𝑧〉)) | 
| 57 |  | df-ov 7434 | . . . . . . . . . . . 12
⊢ (𝑦(Hom ‘𝐷)𝑧) = ((Hom ‘𝐷)‘〈𝑦, 𝑧〉) | 
| 58 | 56, 57 | eqtr4di 2795 | . . . . . . . . . . 11
⊢ (𝑢 = 〈𝑦, 𝑧〉 → ((Hom ‘𝐷)‘𝑢) = (𝑦(Hom ‘𝐷)𝑧)) | 
| 59 | 58 | reseq2d 5997 | . . . . . . . . . 10
⊢ (𝑢 = 〈𝑦, 𝑧〉 → ( I ↾ ((Hom ‘𝐷)‘𝑢)) = ( I ↾ (𝑦(Hom ‘𝐷)𝑧))) | 
| 60 | 59 | mpompt 7547 | . . . . . . . . 9
⊢ (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom
‘𝐷)‘𝑢))) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ ( I ↾ (𝑦(Hom ‘𝐷)𝑧))) | 
| 61 | 55, 60 | eqtr4di 2795 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓))) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom ‘𝐷)‘𝑢)))) | 
| 62 | 22, 61 | opeq12d 4881 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 〈(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓)))〉 = 〈( I ↾
(Base‘𝐷)), (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom
‘𝐷)‘𝑢)))〉) | 
| 63 |  | eqid 2737 | . . . . . . . 8
⊢
(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)) = (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) | 
| 64 | 9 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat) | 
| 65 | 2, 7, 9, 11 | 2ndfcl 18243 | . . . . . . . . 9
⊢ (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷)) | 
| 66 | 65 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷)) | 
| 67 |  | eqid 2737 | . . . . . . . 8
⊢
((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))‘𝑥) = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))‘𝑥) | 
| 68 | 63, 3, 35, 64, 66, 4, 36, 67, 41, 34 | curf1 18270 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥) = 〈(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘(𝐶 2ndF 𝐷))𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑓 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑥, 𝑧〉)𝑓)))〉) | 
| 69 |  | eqid 2737 | . . . . . . . 8
⊢
(idfunc‘𝐷) = (idfunc‘𝐷) | 
| 70 | 69, 4, 64, 41 | idfuval 17921 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) →
(idfunc‘𝐷) = 〈( I ↾ (Base‘𝐷)), (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)) ↦ ( I ↾ ((Hom ‘𝐷)‘𝑢)))〉) | 
| 71 | 62, 68, 70 | 3eqtr4d 2787 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥) = (idfunc‘𝐷)) | 
| 72 |  | eqid 2737 | . . . . . . 7
⊢ (𝑄Δfunc𝐶) = (𝑄Δfunc𝐶) | 
| 73 |  | curf2ndf.q | . . . . . . . . 9
⊢ 𝑄 = (𝐷 FuncCat 𝐷) | 
| 74 | 73, 9, 9 | fuccat 18018 | . . . . . . . 8
⊢ (𝜑 → 𝑄 ∈ Cat) | 
| 75 | 74 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑄 ∈ Cat) | 
| 76 | 73 | fucbas 18008 | . . . . . . 7
⊢ (𝐷 Func 𝐷) = (Base‘𝑄) | 
| 77 | 69 | idfucl 17926 | . . . . . . . . 9
⊢ (𝐷 ∈ Cat →
(idfunc‘𝐷) ∈ (𝐷 Func 𝐷)) | 
| 78 | 9, 77 | syl 17 | . . . . . . . 8
⊢ (𝜑 →
(idfunc‘𝐷) ∈ (𝐷 Func 𝐷)) | 
| 79 | 78 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) →
(idfunc‘𝐷) ∈ (𝐷 Func 𝐷)) | 
| 80 |  | eqid 2737 | . . . . . . 7
⊢
((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) = ((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) | 
| 81 | 72, 75, 35, 76, 79, 80, 3, 36 | diag11 18288 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑥) = (idfunc‘𝐷)) | 
| 82 | 71, 81 | eqtr4d 2780 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥) = ((1st ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑥)) | 
| 83 | 82 | mpteq2dva 5242 | . . . 4
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑥))) | 
| 84 |  | relfunc 17907 | . . . . . . 7
⊢ Rel
(𝐶 Func 𝑄) | 
| 85 | 63, 73, 7, 9, 65 | curfcl 18277 | . . . . . . 7
⊢ (𝜑 → (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) ∈ (𝐶 Func 𝑄)) | 
| 86 |  | 1st2ndbr 8067 | . . . . . . 7
⊢ ((Rel
(𝐶 Func 𝑄) ∧ (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) ∈ (𝐶 Func 𝑄)) → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))(𝐶 Func 𝑄)(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))) | 
| 87 | 84, 85, 86 | sylancr 587 | . . . . . 6
⊢ (𝜑 → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))(𝐶 Func 𝑄)(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))) | 
| 88 | 3, 76, 87 | funcf1 17911 | . . . . 5
⊢ (𝜑 → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))):(Base‘𝐶)⟶(𝐷 Func 𝐷)) | 
| 89 | 88 | feqmptd 6977 | . . . 4
⊢ (𝜑 → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥))) | 
| 90 | 72, 74, 7, 76, 78, 80 | diag1cl 18287 | . . . . . . 7
⊢ (𝜑 → ((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) ∈ (𝐶 Func 𝑄)) | 
| 91 |  | 1st2ndbr 8067 | . . . . . . 7
⊢ ((Rel
(𝐶 Func 𝑄) ∧ ((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) ∈ (𝐶 Func 𝑄)) → (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))(𝐶 Func 𝑄)(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))) | 
| 92 | 84, 90, 91 | sylancr 587 | . . . . . 6
⊢ (𝜑 → (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))(𝐶 Func 𝑄)(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))) | 
| 93 | 3, 76, 92 | funcf1 17911 | . . . . 5
⊢ (𝜑 → (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))):(Base‘𝐶)⟶(𝐷 Func 𝐷)) | 
| 94 | 93 | feqmptd 6977 | . . . 4
⊢ (𝜑 → (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑥))) | 
| 95 | 83, 89, 94 | 3eqtr4d 2787 | . . 3
⊢ (𝜑 → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) = (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))) | 
| 96 | 9 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat) | 
| 97 | 69, 4, 96 | idfu1st 17924 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st
‘(idfunc‘𝐷)) = ( I ↾ (Base‘𝐷))) | 
| 98 | 97 | coeq2d 5873 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((Id‘𝐷) ∘ (1st
‘(idfunc‘𝐷))) = ((Id‘𝐷) ∘ ( I ↾ (Base‘𝐷)))) | 
| 99 |  | eqid 2737 | . . . . . . . . . . 11
⊢
(Id‘𝑄) =
(Id‘𝑄) | 
| 100 |  | eqid 2737 | . . . . . . . . . . 11
⊢
(Id‘𝐷) =
(Id‘𝐷) | 
| 101 | 78 | ad2antrr 726 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) →
(idfunc‘𝐷) ∈ (𝐷 Func 𝐷)) | 
| 102 | 73, 99, 100, 101 | fucid 18019 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((Id‘𝑄)‘(idfunc‘𝐷)) = ((Id‘𝐷) ∘ (1st
‘(idfunc‘𝐷)))) | 
| 103 | 4, 100 | cidfn 17722 | . . . . . . . . . . . . . 14
⊢ (𝐷 ∈ Cat →
(Id‘𝐷) Fn
(Base‘𝐷)) | 
| 104 | 96, 103 | syl 17 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Id‘𝐷) Fn (Base‘𝐷)) | 
| 105 |  | dffn2 6738 | . . . . . . . . . . . . 13
⊢
((Id‘𝐷) Fn
(Base‘𝐷) ↔
(Id‘𝐷):(Base‘𝐷)⟶V) | 
| 106 | 104, 105 | sylib 218 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Id‘𝐷):(Base‘𝐷)⟶V) | 
| 107 | 106 | feqmptd 6977 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Id‘𝐷) = (𝑧 ∈ (Base‘𝐷) ↦ ((Id‘𝐷)‘𝑧))) | 
| 108 |  | fcoi1 6782 | . . . . . . . . . . . 12
⊢
((Id‘𝐷):(Base‘𝐷)⟶V → ((Id‘𝐷) ∘ ( I ↾
(Base‘𝐷))) =
(Id‘𝐷)) | 
| 109 | 106, 108 | syl 17 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((Id‘𝐷) ∘ ( I ↾ (Base‘𝐷))) = (Id‘𝐷)) | 
| 110 | 7 | ad2antrr 726 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐶 ∈ Cat) | 
| 111 | 110 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat) | 
| 112 | 96 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat) | 
| 113 |  | simplrl 777 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶)) | 
| 114 |  | opelxpi 5722 | . . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑥, 𝑧〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) | 
| 115 | 113, 114 | sylan 580 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑥, 𝑧〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) | 
| 116 |  | simplrr 778 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶)) | 
| 117 |  | opelxpi 5722 | . . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑦, 𝑧〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) | 
| 118 | 116, 117 | sylan 580 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑦, 𝑧〉 ∈ ((Base‘𝐶) × (Base‘𝐷))) | 
| 119 | 2, 5, 6, 111, 112, 11, 115, 118 | 2ndf2 18241 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉) = (2nd ↾
(〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c
𝐷))〈𝑦, 𝑧〉))) | 
| 120 | 119 | oveqd 7448 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧)) = (𝑓(2nd ↾ (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉))((Id‘𝐷)‘𝑧))) | 
| 121 |  | df-ov 7434 | . . . . . . . . . . . . . . 15
⊢ (𝑓(2nd ↾
(〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c
𝐷))〈𝑦, 𝑧〉))((Id‘𝐷)‘𝑧)) = ((2nd ↾ (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉))‘〈𝑓, ((Id‘𝐷)‘𝑧)〉) | 
| 122 |  | simplr 769 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) | 
| 123 |  | simpr 484 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑧 ∈ (Base‘𝐷)) | 
| 124 | 4, 41, 100, 112, 123 | catidcl 17725 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((Id‘𝐷)‘𝑧) ∈ (𝑧(Hom ‘𝐷)𝑧)) | 
| 125 | 122, 124 | opelxpd 5724 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑓, ((Id‘𝐷)‘𝑧)〉 ∈ ((𝑥(Hom ‘𝐶)𝑦) × (𝑧(Hom ‘𝐷)𝑧))) | 
| 126 | 113 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑥 ∈ (Base‘𝐶)) | 
| 127 | 116 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐶)) | 
| 128 | 2, 3, 4, 33, 41, 126, 123, 127, 123, 6 | xpchom2 18231 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉) = ((𝑥(Hom ‘𝐶)𝑦) × (𝑧(Hom ‘𝐷)𝑧))) | 
| 129 | 125, 128 | eleqtrrd 2844 | . . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → 〈𝑓, ((Id‘𝐷)‘𝑧)〉 ∈ (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉)) | 
| 130 | 129 | fvresd 6926 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → ((2nd ↾
(〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c
𝐷))〈𝑦, 𝑧〉))‘〈𝑓, ((Id‘𝐷)‘𝑧)〉) = (2nd ‘〈𝑓, ((Id‘𝐷)‘𝑧)〉)) | 
| 131 | 121, 130 | eqtrid 2789 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(2nd ↾ (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉))((Id‘𝐷)‘𝑧)) = (2nd ‘〈𝑓, ((Id‘𝐷)‘𝑧)〉)) | 
| 132 |  | fvex 6919 | . . . . . . . . . . . . . . 15
⊢
((Id‘𝐷)‘𝑧) ∈ V | 
| 133 | 47, 132 | op2nd 8023 | . . . . . . . . . . . . . 14
⊢
(2nd ‘〈𝑓, ((Id‘𝐷)‘𝑧)〉) = ((Id‘𝐷)‘𝑧) | 
| 134 | 131, 133 | eqtrdi 2793 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(2nd ↾ (〈𝑥, 𝑧〉(Hom ‘(𝐶 ×c 𝐷))〈𝑦, 𝑧〉))((Id‘𝐷)‘𝑧)) = ((Id‘𝐷)‘𝑧)) | 
| 135 | 120, 134 | eqtrd 2777 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑧 ∈ (Base‘𝐷)) → (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧)) = ((Id‘𝐷)‘𝑧)) | 
| 136 | 135 | mpteq2dva 5242 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧))) = (𝑧 ∈ (Base‘𝐷) ↦ ((Id‘𝐷)‘𝑧))) | 
| 137 | 107, 109,
136 | 3eqtr4rd 2788 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧))) = ((Id‘𝐷) ∘ ( I ↾ (Base‘𝐷)))) | 
| 138 | 98, 102, 137 | 3eqtr4rd 2788 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧))) = ((Id‘𝑄)‘(idfunc‘𝐷))) | 
| 139 | 65 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷)) | 
| 140 |  | simpr 484 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) | 
| 141 |  | eqid 2737 | . . . . . . . . . 10
⊢ ((𝑥(2nd
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))𝑦)‘𝑓) = ((𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)‘𝑓) | 
| 142 | 63, 3, 110, 96, 139, 4, 33, 100, 113, 116, 140, 141 | curf2 18274 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)‘𝑓) = (𝑧 ∈ (Base‘𝐷) ↦ (𝑓(〈𝑥, 𝑧〉(2nd ‘(𝐶
2ndF 𝐷))〈𝑦, 𝑧〉)((Id‘𝐷)‘𝑧)))) | 
| 143 | 74 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑄 ∈ Cat) | 
| 144 | 72, 143, 110, 76, 101, 80, 3, 113, 33, 99, 116, 140 | diag12 18289 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)‘𝑓) = ((Id‘𝑄)‘(idfunc‘𝐷))) | 
| 145 | 138, 142,
144 | 3eqtr4d 2787 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)‘𝑓) = ((𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)‘𝑓)) | 
| 146 | 145 | mpteq2dva 5242 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)‘𝑓)) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)‘𝑓))) | 
| 147 |  | eqid 2737 | . . . . . . . . . 10
⊢ (𝐷 Nat 𝐷) = (𝐷 Nat 𝐷) | 
| 148 | 73, 147 | fuchom 18009 | . . . . . . . . 9
⊢ (𝐷 Nat 𝐷) = (Hom ‘𝑄) | 
| 149 | 87 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))(𝐶 Func 𝑄)(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))) | 
| 150 |  | simprl 771 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | 
| 151 |  | simprr 773 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | 
| 152 | 3, 33, 148, 149, 150, 151 | funcf2 17913 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷)))‘𝑥)(𝐷 Nat 𝐷)((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))‘𝑦))) | 
| 153 | 152 | feqmptd 6977 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)‘𝑓))) | 
| 154 | 92 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))(𝐶 Func 𝑄)(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))) | 
| 155 | 3, 33, 148, 154, 150, 151 | funcf2 17913 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑥)(𝐷 Nat 𝐷)((1st ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))‘𝑦))) | 
| 156 | 155 | feqmptd 6977 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)‘𝑓))) | 
| 157 | 146, 153,
156 | 3eqtr4d 2787 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦) = (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)) | 
| 158 | 157 | 3impb 1115 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦) = (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦)) | 
| 159 | 158 | mpoeq3dva 7510 | . . . 4
⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦))) | 
| 160 | 3, 87 | funcfn2 17914 | . . . . 5
⊢ (𝜑 → (2nd
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶))) | 
| 161 |  | fnov 7564 | . . . . 5
⊢
((2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦))) | 
| 162 | 160, 161 | sylib 218 | . . . 4
⊢ (𝜑 → (2nd
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))𝑦))) | 
| 163 | 3, 92 | funcfn2 17914 | . . . . 5
⊢ (𝜑 → (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶))) | 
| 164 |  | fnov 7564 | . . . . 5
⊢
((2nd ‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦))) | 
| 165 | 163, 164 | sylib 218 | . . . 4
⊢ (𝜑 → (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))𝑦))) | 
| 166 | 159, 162,
165 | 3eqtr4d 2787 | . . 3
⊢ (𝜑 → (2nd
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))) = (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))) | 
| 167 | 95, 166 | opeq12d 4881 | . 2
⊢ (𝜑 → 〈(1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))), (2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))〉 = 〈(1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))), (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))〉) | 
| 168 |  | 1st2nd 8064 | . . 3
⊢ ((Rel
(𝐶 Func 𝑄) ∧ (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) ∈ (𝐶 Func 𝑄)) → (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) = 〈(1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))), (2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))〉) | 
| 169 | 84, 85, 168 | sylancr 587 | . 2
⊢ (𝜑 → (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) = 〈(1st
‘(〈𝐶, 𝐷〉 curryF
(𝐶
2ndF 𝐷))), (2nd ‘(〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)))〉) | 
| 170 |  | 1st2nd 8064 | . . 3
⊢ ((Rel
(𝐶 Func 𝑄) ∧ ((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) ∈ (𝐶 Func 𝑄)) → ((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) = 〈(1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))), (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))〉) | 
| 171 | 84, 90, 170 | sylancr 587 | . 2
⊢ (𝜑 → ((1st
‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)) = 〈(1st
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))), (2nd
‘((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))〉) | 
| 172 | 167, 169,
171 | 3eqtr4d 2787 | 1
⊢ (𝜑 → (〈𝐶, 𝐷〉 curryF (𝐶
2ndF 𝐷)) = ((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) |