Step | Hyp | Ref
| Expression |
1 | | 1stfcl.t |
. . . 4
⊢ 𝑇 = (𝐶 ×c 𝐷) |
2 | | eqid 2825 |
. . . . 5
⊢
(Base‘𝐶) =
(Base‘𝐶) |
3 | | eqid 2825 |
. . . . 5
⊢
(Base‘𝐷) =
(Base‘𝐷) |
4 | 1, 2, 3 | xpcbas 17178 |
. . . 4
⊢
((Base‘𝐶)
× (Base‘𝐷)) =
(Base‘𝑇) |
5 | | eqid 2825 |
. . . 4
⊢ (Hom
‘𝑇) = (Hom
‘𝑇) |
6 | | 1stfcl.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | | 1stfcl.d |
. . . 4
⊢ (𝜑 → 𝐷 ∈ Cat) |
8 | | 2ndfcl.p |
. . . 4
⊢ 𝑄 = (𝐶 2ndF 𝐷) |
9 | 1, 4, 5, 6, 7, 8 | 2ndfval 17194 |
. . 3
⊢ (𝜑 → 𝑄 = 〈(2nd ↾
((Base‘𝐶) ×
(Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))〉) |
10 | | fo2nd 7454 |
. . . . . . . 8
⊢
2nd :V–onto→V |
11 | | fofun 6358 |
. . . . . . . 8
⊢
(2nd :V–onto→V → Fun 2nd ) |
12 | 10, 11 | ax-mp 5 |
. . . . . . 7
⊢ Fun
2nd |
13 | | fvex 6450 |
. . . . . . . 8
⊢
(Base‘𝐶)
∈ V |
14 | | fvex 6450 |
. . . . . . . 8
⊢
(Base‘𝐷)
∈ V |
15 | 13, 14 | xpex 7228 |
. . . . . . 7
⊢
((Base‘𝐶)
× (Base‘𝐷))
∈ V |
16 | | resfunexg 6740 |
. . . . . . 7
⊢ ((Fun
2nd ∧ ((Base‘𝐶) × (Base‘𝐷)) ∈ V) → (2nd ↾
((Base‘𝐶) ×
(Base‘𝐷))) ∈
V) |
17 | 12, 15, 16 | mp2an 683 |
. . . . . 6
⊢
(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V |
18 | 15, 15 | mpt2ex 7515 |
. . . . . 6
⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) ∈ V |
19 | 17, 18 | op2ndd 7444 |
. . . . 5
⊢ (𝑄 = 〈(2nd ↾
((Base‘𝐶) ×
(Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))〉 → (2nd ‘𝑄) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))) |
20 | 9, 19 | syl 17 |
. . . 4
⊢ (𝜑 → (2nd
‘𝑄) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))) |
21 | 20 | opeq2d 4632 |
. . 3
⊢ (𝜑 → 〈(2nd
↾ ((Base‘𝐶)
× (Base‘𝐷))),
(2nd ‘𝑄)〉 = 〈(2nd ↾
((Base‘𝐶) ×
(Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))〉) |
22 | 9, 21 | eqtr4d 2864 |
. 2
⊢ (𝜑 → 𝑄 = 〈(2nd ↾
((Base‘𝐶) ×
(Base‘𝐷))),
(2nd ‘𝑄)〉) |
23 | | eqid 2825 |
. . . 4
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) |
24 | | eqid 2825 |
. . . 4
⊢
(Id‘𝑇) =
(Id‘𝑇) |
25 | | eqid 2825 |
. . . 4
⊢
(Id‘𝐷) =
(Id‘𝐷) |
26 | | eqid 2825 |
. . . 4
⊢
(comp‘𝑇) =
(comp‘𝑇) |
27 | | eqid 2825 |
. . . 4
⊢
(comp‘𝐷) =
(comp‘𝐷) |
28 | 1, 6, 7 | xpccat 17190 |
. . . 4
⊢ (𝜑 → 𝑇 ∈ Cat) |
29 | | f2ndres 7458 |
. . . . 5
⊢
(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐷) |
30 | 29 | a1i 11 |
. . . 4
⊢ (𝜑 → (2nd ↾
((Base‘𝐶) ×
(Base‘𝐷))):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐷)) |
31 | | eqid 2825 |
. . . . . 6
⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) |
32 | | ovex 6942 |
. . . . . . 7
⊢ (𝑥(Hom ‘𝑇)𝑦) ∈ V |
33 | | resfunexg 6740 |
. . . . . . 7
⊢ ((Fun
2nd ∧ (𝑥(Hom
‘𝑇)𝑦) ∈ V) → (2nd ↾
(𝑥(Hom ‘𝑇)𝑦)) ∈ V) |
34 | 12, 32, 33 | mp2an 683 |
. . . . . 6
⊢
(2nd ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V |
35 | 31, 34 | fnmpt2i 7507 |
. . . . 5
⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))) |
36 | 20 | fneq1d 6218 |
. . . . 5
⊢ (𝜑 → ((2nd
‘𝑄) Fn
(((Base‘𝐶) ×
(Base‘𝐷)) ×
((Base‘𝐶) ×
(Base‘𝐷))) ↔
(𝑥 ∈
((Base‘𝐶) ×
(Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd
↾ (𝑥(Hom ‘𝑇)𝑦))) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))) |
37 | 35, 36 | mpbiri 250 |
. . . 4
⊢ (𝜑 → (2nd
‘𝑄) Fn
(((Base‘𝐶) ×
(Base‘𝐷)) ×
((Base‘𝐶) ×
(Base‘𝐷)))) |
38 | | f2ndres 7458 |
. . . . . 6
⊢
(2nd ↾ (((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))):(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))⟶((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)) |
39 | 6 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐶 ∈ Cat) |
40 | 7 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐷 ∈ Cat) |
41 | | simprl 787 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
42 | | simprr 789 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
43 | 1, 4, 5, 39, 40, 8, 41, 42 | 2ndf2 17196 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd ‘𝑄)𝑦) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) |
44 | | eqid 2825 |
. . . . . . . . . 10
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
45 | 1, 4, 44, 23, 5, 41, 42 | xpchom 17180 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(Hom ‘𝑇)𝑦) = (((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))) |
46 | 45 | reseq2d 5633 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)) = (2nd ↾
(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦))))) |
47 | 43, 46 | eqtrd 2861 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd ‘𝑄)𝑦) = (2nd ↾ (((1st
‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd
‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦))))) |
48 | 47 | feq1d 6267 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((𝑥(2nd ‘𝑄)𝑦):(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))⟶((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)) ↔ (2nd ↾
(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))):(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))⟶((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))) |
49 | 38, 48 | mpbiri 250 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd ‘𝑄)𝑦):(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))⟶((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦))) |
50 | | fvres 6456 |
. . . . . . . 8
⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((2nd
↾ ((Base‘𝐶)
× (Base‘𝐷)))‘𝑥) = (2nd ‘𝑥)) |
51 | 50 | ad2antrl 719 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥) = (2nd ‘𝑥)) |
52 | | fvres 6456 |
. . . . . . . 8
⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((2nd
↾ ((Base‘𝐶)
× (Base‘𝐷)))‘𝑦) = (2nd ‘𝑦)) |
53 | 52 | ad2antll 720 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑦) = (2nd ‘𝑦)) |
54 | 51, 53 | oveq12d 6928 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥)(Hom ‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)) = ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦))) |
55 | 45, 54 | feq23d 6277 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((𝑥(2nd ‘𝑄)𝑦):(𝑥(Hom ‘𝑇)𝑦)⟶(((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥)(Hom ‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)) ↔ (𝑥(2nd ‘𝑄)𝑦):(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))⟶((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))) |
56 | 49, 55 | mpbird 249 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd ‘𝑄)𝑦):(𝑥(Hom ‘𝑇)𝑦)⟶(((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥)(Hom ‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦))) |
57 | 28 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑇 ∈ Cat) |
58 | | simpr 479 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
59 | 4, 5, 24, 57, 58 | catidcl 16702 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥)) |
60 | | fvres 6456 |
. . . . . . 7
⊢
(((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (2nd ‘((Id‘𝑇)‘𝑥))) |
61 | 59, 60 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (2nd ‘((Id‘𝑇)‘𝑥))) |
62 | | 1st2nd2 7472 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) |
63 | 62 | adantl 475 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) |
64 | 63 | fveq2d 6441 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ((Id‘𝑇)‘〈(1st ‘𝑥), (2nd ‘𝑥)〉)) |
65 | 6 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐶 ∈ Cat) |
66 | 7 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐷 ∈ Cat) |
67 | | eqid 2825 |
. . . . . . . . 9
⊢
(Id‘𝐶) =
(Id‘𝐶) |
68 | | xp1st 7465 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1st
‘𝑥) ∈
(Base‘𝐶)) |
69 | 68 | adantl 475 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1st ‘𝑥) ∈ (Base‘𝐶)) |
70 | | xp2nd 7466 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2nd
‘𝑥) ∈
(Base‘𝐷)) |
71 | 70 | adantl 475 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2nd ‘𝑥) ∈ (Base‘𝐷)) |
72 | 1, 65, 66, 2, 3, 67, 25, 24, 69, 71 | xpcid 17189 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘〈(1st ‘𝑥), (2nd ‘𝑥)〉) =
〈((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐷)‘(2nd ‘𝑥))〉) |
73 | 64, 72 | eqtrd 2861 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = 〈((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐷)‘(2nd ‘𝑥))〉) |
74 | | fvex 6450 |
. . . . . . . 8
⊢
((Id‘𝐶)‘(1st ‘𝑥)) ∈ V |
75 | | fvex 6450 |
. . . . . . . 8
⊢
((Id‘𝐷)‘(2nd ‘𝑥)) ∈ V |
76 | 74, 75 | op2ndd 7444 |
. . . . . . 7
⊢
(((Id‘𝑇)‘𝑥) = 〈((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐷)‘(2nd ‘𝑥))〉 → (2nd
‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘(2nd ‘𝑥))) |
77 | 73, 76 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2nd
‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘(2nd ‘𝑥))) |
78 | 61, 77 | eqtrd 2861 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘(2nd ‘𝑥))) |
79 | 1, 4, 5, 65, 66, 8, 58, 58 | 2ndf2 17196 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(2nd ‘𝑄)𝑥) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑥))) |
80 | 79 | fveq1d 6439 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2nd ‘𝑄)𝑥)‘((Id‘𝑇)‘𝑥)) = ((2nd ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥))) |
81 | 50 | adantl 475 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥) = (2nd ‘𝑥)) |
82 | 81 | fveq2d 6441 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝐷)‘((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥)) = ((Id‘𝐷)‘(2nd ‘𝑥))) |
83 | 78, 80, 82 | 3eqtr4d 2871 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2nd ‘𝑄)𝑥)‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥))) |
84 | 28 | 3ad2ant1 1167 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑇 ∈ Cat) |
85 | | simp21 1267 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
86 | | simp22 1268 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
87 | | simp23 1269 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) |
88 | | simp3l 1262 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦)) |
89 | | simp3r 1263 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧)) |
90 | 4, 5, 26, 84, 85, 86, 87, 88, 89 | catcocl 16705 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑔(〈𝑥, 𝑦〉(comp‘𝑇)𝑧)𝑓) ∈ (𝑥(Hom ‘𝑇)𝑧)) |
91 | | fvres 6456 |
. . . . . . 7
⊢ ((𝑔(〈𝑥, 𝑦〉(comp‘𝑇)𝑧)𝑓) ∈ (𝑥(Hom ‘𝑇)𝑧) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(〈𝑥, 𝑦〉(comp‘𝑇)𝑧)𝑓)) = (2nd ‘(𝑔(〈𝑥, 𝑦〉(comp‘𝑇)𝑧)𝑓))) |
92 | 90, 91 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(〈𝑥, 𝑦〉(comp‘𝑇)𝑧)𝑓)) = (2nd ‘(𝑔(〈𝑥, 𝑦〉(comp‘𝑇)𝑧)𝑓))) |
93 | 1, 4, 5, 26, 85, 86, 87, 88, 89, 27 | xpcco2nd 17185 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (2nd ‘(𝑔(〈𝑥, 𝑦〉(comp‘𝑇)𝑧)𝑓)) = ((2nd ‘𝑔)(〈(2nd
‘𝑥), (2nd
‘𝑦)〉(comp‘𝐷)(2nd ‘𝑧))(2nd ‘𝑓))) |
94 | 92, 93 | eqtrd 2861 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(〈𝑥, 𝑦〉(comp‘𝑇)𝑧)𝑓)) = ((2nd ‘𝑔)(〈(2nd
‘𝑥), (2nd
‘𝑦)〉(comp‘𝐷)(2nd ‘𝑧))(2nd ‘𝑓))) |
95 | 6 | 3ad2ant1 1167 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝐶 ∈ Cat) |
96 | 7 | 3ad2ant1 1167 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝐷 ∈ Cat) |
97 | 1, 4, 5, 95, 96, 8, 85, 87 | 2ndf2 17196 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑥(2nd ‘𝑄)𝑧) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑧))) |
98 | 97 | fveq1d 6439 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2nd ‘𝑄)𝑧)‘(𝑔(〈𝑥, 𝑦〉(comp‘𝑇)𝑧)𝑓)) = ((2nd ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(〈𝑥, 𝑦〉(comp‘𝑇)𝑧)𝑓))) |
99 | 85, 50 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥) = (2nd ‘𝑥)) |
100 | 86, 52 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑦) = (2nd ‘𝑦)) |
101 | 99, 100 | opeq12d 4633 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 〈((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥), ((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑦)〉 = 〈(2nd ‘𝑥), (2nd ‘𝑦)〉) |
102 | | fvres 6456 |
. . . . . . . 8
⊢ (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((2nd
↾ ((Base‘𝐶)
× (Base‘𝐷)))‘𝑧) = (2nd ‘𝑧)) |
103 | 87, 102 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑧) = (2nd ‘𝑧)) |
104 | 101, 103 | oveq12d 6928 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (〈((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥), ((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑦)〉(comp‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧)) = (〈(2nd ‘𝑥), (2nd ‘𝑦)〉(comp‘𝐷)(2nd ‘𝑧))) |
105 | 1, 4, 5, 95, 96, 8, 86, 87 | 2ndf2 17196 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑦(2nd ‘𝑄)𝑧) = (2nd ↾ (𝑦(Hom ‘𝑇)𝑧))) |
106 | 105 | fveq1d 6439 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑦(2nd ‘𝑄)𝑧)‘𝑔) = ((2nd ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔)) |
107 | | fvres 6456 |
. . . . . . . 8
⊢ (𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧) → ((2nd ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔) = (2nd ‘𝑔)) |
108 | 89, 107 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔) = (2nd ‘𝑔)) |
109 | 106, 108 | eqtrd 2861 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑦(2nd ‘𝑄)𝑧)‘𝑔) = (2nd ‘𝑔)) |
110 | 1, 4, 5, 95, 96, 8, 85, 86 | 2ndf2 17196 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑥(2nd ‘𝑄)𝑦) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) |
111 | 110 | fveq1d 6439 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2nd ‘𝑄)𝑦)‘𝑓) = ((2nd ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓)) |
112 | | fvres 6456 |
. . . . . . . 8
⊢ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓) = (2nd ‘𝑓)) |
113 | 88, 112 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓) = (2nd ‘𝑓)) |
114 | 111, 113 | eqtrd 2861 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2nd ‘𝑄)𝑦)‘𝑓) = (2nd ‘𝑓)) |
115 | 104, 109,
114 | oveq123d 6931 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (((𝑦(2nd ‘𝑄)𝑧)‘𝑔)(〈((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥), ((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑦)〉(comp‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧))((𝑥(2nd ‘𝑄)𝑦)‘𝑓)) = ((2nd ‘𝑔)(〈(2nd
‘𝑥), (2nd
‘𝑦)〉(comp‘𝐷)(2nd ‘𝑧))(2nd ‘𝑓))) |
116 | 94, 98, 115 | 3eqtr4d 2871 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2nd ‘𝑄)𝑧)‘(𝑔(〈𝑥, 𝑦〉(comp‘𝑇)𝑧)𝑓)) = (((𝑦(2nd ‘𝑄)𝑧)‘𝑔)(〈((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑥), ((2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))‘𝑦)〉(comp‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧))((𝑥(2nd ‘𝑄)𝑦)‘𝑓))) |
117 | 4, 3, 5, 23, 24, 25, 26, 27, 28, 7, 30, 37, 56, 83, 116 | isfuncd 16884 |
. . 3
⊢ (𝜑 → (2nd ↾
((Base‘𝐶) ×
(Base‘𝐷)))(𝑇 Func 𝐷)(2nd ‘𝑄)) |
118 | | df-br 4876 |
. . 3
⊢
((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐷)(2nd ‘𝑄) ↔ 〈(2nd ↾
((Base‘𝐶) ×
(Base‘𝐷))),
(2nd ‘𝑄)〉 ∈ (𝑇 Func 𝐷)) |
119 | 117, 118 | sylib 210 |
. 2
⊢ (𝜑 → 〈(2nd
↾ ((Base‘𝐶)
× (Base‘𝐷))),
(2nd ‘𝑄)〉 ∈ (𝑇 Func 𝐷)) |
120 | 22, 119 | eqeltrd 2906 |
1
⊢ (𝜑 → 𝑄 ∈ (𝑇 Func 𝐷)) |