Metamath Proof Explorer


Theorem curf1cl

Description: The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-Jan-2017)

Ref Expression
Hypotheses curfval.g 𝐺 = ( ⟨ 𝐶 , 𝐷 ⟩ curryF 𝐹 )
curfval.a 𝐴 = ( Base ‘ 𝐶 )
curfval.c ( 𝜑𝐶 ∈ Cat )
curfval.d ( 𝜑𝐷 ∈ Cat )
curfval.f ( 𝜑𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) )
curfval.b 𝐵 = ( Base ‘ 𝐷 )
curf1.x ( 𝜑𝑋𝐴 )
curf1.k 𝐾 = ( ( 1st𝐺 ) ‘ 𝑋 )
Assertion curf1cl ( 𝜑𝐾 ∈ ( 𝐷 Func 𝐸 ) )

Proof

Step Hyp Ref Expression
1 curfval.g 𝐺 = ( ⟨ 𝐶 , 𝐷 ⟩ curryF 𝐹 )
2 curfval.a 𝐴 = ( Base ‘ 𝐶 )
3 curfval.c ( 𝜑𝐶 ∈ Cat )
4 curfval.d ( 𝜑𝐷 ∈ Cat )
5 curfval.f ( 𝜑𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) )
6 curfval.b 𝐵 = ( Base ‘ 𝐷 )
7 curf1.x ( 𝜑𝑋𝐴 )
8 curf1.k 𝐾 = ( ( 1st𝐺 ) ‘ 𝑋 )
9 eqid ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
10 eqid ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 )
11 1 2 3 4 5 6 7 8 9 10 curf1 ( 𝜑𝐾 = ⟨ ( 𝑦𝐵 ↦ ( 𝑋 ( 1st𝐹 ) 𝑦 ) ) , ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) ⟩ )
12 6 fvexi 𝐵 ∈ V
13 12 mptex ( 𝑦𝐵 ↦ ( 𝑋 ( 1st𝐹 ) 𝑦 ) ) ∈ V
14 12 12 mpoex ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) ∈ V
15 13 14 op1std ( 𝐾 = ⟨ ( 𝑦𝐵 ↦ ( 𝑋 ( 1st𝐹 ) 𝑦 ) ) , ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) ⟩ → ( 1st𝐾 ) = ( 𝑦𝐵 ↦ ( 𝑋 ( 1st𝐹 ) 𝑦 ) ) )
16 11 15 syl ( 𝜑 → ( 1st𝐾 ) = ( 𝑦𝐵 ↦ ( 𝑋 ( 1st𝐹 ) 𝑦 ) ) )
17 13 14 op2ndd ( 𝐾 = ⟨ ( 𝑦𝐵 ↦ ( 𝑋 ( 1st𝐹 ) 𝑦 ) ) , ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) ⟩ → ( 2nd𝐾 ) = ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) )
18 11 17 syl ( 𝜑 → ( 2nd𝐾 ) = ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) )
19 16 18 opeq12d ( 𝜑 → ⟨ ( 1st𝐾 ) , ( 2nd𝐾 ) ⟩ = ⟨ ( 𝑦𝐵 ↦ ( 𝑋 ( 1st𝐹 ) 𝑦 ) ) , ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) ⟩ )
20 11 19 eqtr4d ( 𝜑𝐾 = ⟨ ( 1st𝐾 ) , ( 2nd𝐾 ) ⟩ )
21 eqid ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 )
22 eqid ( Hom ‘ 𝐸 ) = ( Hom ‘ 𝐸 )
23 eqid ( Id ‘ 𝐷 ) = ( Id ‘ 𝐷 )
24 eqid ( Id ‘ 𝐸 ) = ( Id ‘ 𝐸 )
25 eqid ( comp ‘ 𝐷 ) = ( comp ‘ 𝐷 )
26 eqid ( comp ‘ 𝐸 ) = ( comp ‘ 𝐸 )
27 funcrcl ( 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) → ( ( 𝐶 ×c 𝐷 ) ∈ Cat ∧ 𝐸 ∈ Cat ) )
28 5 27 syl ( 𝜑 → ( ( 𝐶 ×c 𝐷 ) ∈ Cat ∧ 𝐸 ∈ Cat ) )
29 28 simprd ( 𝜑𝐸 ∈ Cat )
30 eqid ( 𝐶 ×c 𝐷 ) = ( 𝐶 ×c 𝐷 )
31 30 2 6 xpcbas ( 𝐴 × 𝐵 ) = ( Base ‘ ( 𝐶 ×c 𝐷 ) )
32 relfunc Rel ( ( 𝐶 ×c 𝐷 ) Func 𝐸 )
33 1st2ndbr ( ( Rel ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ∧ 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ) → ( 1st𝐹 ) ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ( 2nd𝐹 ) )
34 32 5 33 sylancr ( 𝜑 → ( 1st𝐹 ) ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ( 2nd𝐹 ) )
35 31 21 34 funcf1 ( 𝜑 → ( 1st𝐹 ) : ( 𝐴 × 𝐵 ) ⟶ ( Base ‘ 𝐸 ) )
36 35 adantr ( ( 𝜑𝑦𝐵 ) → ( 1st𝐹 ) : ( 𝐴 × 𝐵 ) ⟶ ( Base ‘ 𝐸 ) )
37 7 adantr ( ( 𝜑𝑦𝐵 ) → 𝑋𝐴 )
38 simpr ( ( 𝜑𝑦𝐵 ) → 𝑦𝐵 )
39 36 37 38 fovrnd ( ( 𝜑𝑦𝐵 ) → ( 𝑋 ( 1st𝐹 ) 𝑦 ) ∈ ( Base ‘ 𝐸 ) )
40 16 39 fmpt3d ( 𝜑 → ( 1st𝐾 ) : 𝐵 ⟶ ( Base ‘ 𝐸 ) )
41 eqid ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) = ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) )
42 ovex ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∈ V
43 42 mptex ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ∈ V
44 41 43 fnmpoi ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) Fn ( 𝐵 × 𝐵 )
45 18 fneq1d ( 𝜑 → ( ( 2nd𝐾 ) Fn ( 𝐵 × 𝐵 ) ↔ ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) Fn ( 𝐵 × 𝐵 ) ) )
46 44 45 mpbiri ( 𝜑 → ( 2nd𝐾 ) Fn ( 𝐵 × 𝐵 ) )
47 18 oveqd ( 𝜑 → ( 𝑦 ( 2nd𝐾 ) 𝑧 ) = ( 𝑦 ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) 𝑧 ) )
48 41 ovmpt4g ( ( 𝑦𝐵𝑧𝐵 ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ∈ V ) → ( 𝑦 ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) 𝑧 ) = ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) )
49 43 48 mp3an3 ( ( 𝑦𝐵𝑧𝐵 ) → ( 𝑦 ( 𝑦𝐵 , 𝑧𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) ) 𝑧 ) = ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) )
50 47 49 sylan9eq ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) → ( 𝑦 ( 2nd𝐾 ) 𝑧 ) = ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ) )
51 eqid ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) = ( Hom ‘ ( 𝐶 ×c 𝐷 ) )
52 34 ad2antrr ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 1st𝐹 ) ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ( 2nd𝐹 ) )
53 7 ad2antrr ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝑋𝐴 )
54 simplrl ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝑦𝐵 )
55 53 54 opelxpd ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ⟨ 𝑋 , 𝑦 ⟩ ∈ ( 𝐴 × 𝐵 ) )
56 simplrr ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝑧𝐵 )
57 53 56 opelxpd ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ⟨ 𝑋 , 𝑧 ⟩ ∈ ( 𝐴 × 𝐵 ) )
58 31 51 22 52 55 57 funcf2 ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) : ( ⟨ 𝑋 , 𝑦 ⟩ ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) ⟨ 𝑋 , 𝑧 ⟩ ) ⟶ ( ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ( Hom ‘ 𝐸 ) ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ) )
59 eqid ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
60 30 31 59 9 51 55 57 xpchom ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 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 ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝐶 ∈ Cat )
62 4 ad2antrr ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝐷 ∈ Cat )
63 5 ad2antrr ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) )
64 1 2 61 62 63 6 53 8 54 curf11 ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 1st𝐾 ) ‘ 𝑦 ) = ( 𝑋 ( 1st𝐹 ) 𝑦 ) )
65 df-ov ( 𝑋 ( 1st𝐹 ) 𝑦 ) = ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ )
66 64 65 eqtr2di ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) = ( ( 1st𝐾 ) ‘ 𝑦 ) )
67 1 2 61 62 63 6 53 8 56 curf11 ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 1st𝐾 ) ‘ 𝑧 ) = ( 𝑋 ( 1st𝐹 ) 𝑧 ) )
68 df-ov ( 𝑋 ( 1st𝐹 ) 𝑧 ) = ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑧 ⟩ )
69 67 68 eqtr2di ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑧 ⟩ ) = ( ( 1st𝐾 ) ‘ 𝑧 ) )
70 66 69 oveq12d ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ( Hom ‘ 𝐸 ) ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ) = ( ( ( 1st𝐾 ) ‘ 𝑦 ) ( Hom ‘ 𝐸 ) ( ( 1st𝐾 ) ‘ 𝑧 ) ) )
71 60 70 feq23d ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) : ( ⟨ 𝑋 , 𝑦 ⟩ ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) ⟨ 𝑋 , 𝑧 ⟩ ) ⟶ ( ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ( Hom ‘ 𝐸 ) ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ) ↔ ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) : ( ( ( 1st ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ( Hom ‘ 𝐶 ) ( 1st ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ) × ( ( 2nd ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ( Hom ‘ 𝐷 ) ( 2nd ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ) ) ⟶ ( ( ( 1st𝐾 ) ‘ 𝑦 ) ( Hom ‘ 𝐸 ) ( ( 1st𝐾 ) ‘ 𝑧 ) ) ) )
72 58 71 mpbid ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) : ( ( ( 1st ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ( Hom ‘ 𝐶 ) ( 1st ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ) × ( ( 2nd ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ( Hom ‘ 𝐷 ) ( 2nd ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ) ) ⟶ ( ( ( 1st𝐾 ) ‘ 𝑦 ) ( Hom ‘ 𝐸 ) ( ( 1st𝐾 ) ‘ 𝑧 ) ) )
73 2 59 10 61 53 catidcl ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) )
74 op1stg ( ( 𝑋𝐴𝑦𝐵 ) → ( 1st ‘ ⟨ 𝑋 , 𝑦 ⟩ ) = 𝑋 )
75 53 54 74 syl2anc ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 1st ‘ ⟨ 𝑋 , 𝑦 ⟩ ) = 𝑋 )
76 op1stg ( ( 𝑋𝐴𝑧𝐵 ) → ( 1st ‘ ⟨ 𝑋 , 𝑧 ⟩ ) = 𝑋 )
77 53 56 76 syl2anc ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 1st ‘ ⟨ 𝑋 , 𝑧 ⟩ ) = 𝑋 )
78 75 77 oveq12d ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 1st ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ( Hom ‘ 𝐶 ) ( 1st ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ) = ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) )
79 73 78 eleqtrrd ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ∈ ( ( 1st ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ( Hom ‘ 𝐶 ) ( 1st ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ) )
80 simpr ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) )
81 op2ndg ( ( 𝑋𝐴𝑦𝐵 ) → ( 2nd ‘ ⟨ 𝑋 , 𝑦 ⟩ ) = 𝑦 )
82 53 54 81 syl2anc ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 2nd ‘ ⟨ 𝑋 , 𝑦 ⟩ ) = 𝑦 )
83 op2ndg ( ( 𝑋𝐴𝑧𝐵 ) → ( 2nd ‘ ⟨ 𝑋 , 𝑧 ⟩ ) = 𝑧 )
84 53 56 83 syl2anc ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 2nd ‘ ⟨ 𝑋 , 𝑧 ⟩ ) = 𝑧 )
85 82 84 oveq12d ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 2nd ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ( Hom ‘ 𝐷 ) ( 2nd ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ) = ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) )
86 80 85 eleqtrrd ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝑔 ∈ ( ( 2nd ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ( Hom ‘ 𝐷 ) ( 2nd ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ) )
87 72 79 86 fovrnd ( ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) ∈ ( ( ( 1st𝐾 ) ‘ 𝑦 ) ( Hom ‘ 𝐸 ) ( ( 1st𝐾 ) ‘ 𝑧 ) ) )
88 50 87 fmpt3d ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵 ) ) → ( 𝑦 ( 2nd𝐾 ) 𝑧 ) : ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ⟶ ( ( ( 1st𝐾 ) ‘ 𝑦 ) ( Hom ‘ 𝐸 ) ( ( 1st𝐾 ) ‘ 𝑧 ) ) )
89 3 adantr ( ( 𝜑𝑦𝐵 ) → 𝐶 ∈ Cat )
90 4 adantr ( ( 𝜑𝑦𝐵 ) → 𝐷 ∈ Cat )
91 eqid ( Id ‘ ( 𝐶 ×c 𝐷 ) ) = ( Id ‘ ( 𝐶 ×c 𝐷 ) )
92 30 89 90 2 6 10 23 91 37 38 xpcid ( ( 𝜑𝑦𝐵 ) → ( ( Id ‘ ( 𝐶 ×c 𝐷 ) ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) = ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ⟩ )
93 92 fveq2d ( ( 𝜑𝑦𝐵 ) → ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑦 ⟩ ) ‘ ( ( Id ‘ ( 𝐶 ×c 𝐷 ) ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ) = ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑦 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ⟩ ) )
94 df-ov ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑦 ⟩ ) ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) = ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑦 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ⟩ )
95 93 94 eqtr4di ( ( 𝜑𝑦𝐵 ) → ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑦 ⟩ ) ‘ ( ( Id ‘ ( 𝐶 ×c 𝐷 ) ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑦 ⟩ ) ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) )
96 34 adantr ( ( 𝜑𝑦𝐵 ) → ( 1st𝐹 ) ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ( 2nd𝐹 ) )
97 opelxpi ( ( 𝑋𝐴𝑦𝐵 ) → ⟨ 𝑋 , 𝑦 ⟩ ∈ ( 𝐴 × 𝐵 ) )
98 7 97 sylan ( ( 𝜑𝑦𝐵 ) → ⟨ 𝑋 , 𝑦 ⟩ ∈ ( 𝐴 × 𝐵 ) )
99 31 91 24 96 98 funcid ( ( 𝜑𝑦𝐵 ) → ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑦 ⟩ ) ‘ ( ( Id ‘ ( 𝐶 ×c 𝐷 ) ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ) = ( ( Id ‘ 𝐸 ) ‘ ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ) )
100 95 99 eqtr3d ( ( 𝜑𝑦𝐵 ) → ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑦 ⟩ ) ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) = ( ( Id ‘ 𝐸 ) ‘ ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ) )
101 5 adantr ( ( 𝜑𝑦𝐵 ) → 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) )
102 6 9 23 90 38 catidcl ( ( 𝜑𝑦𝐵 ) → ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑦 ) )
103 1 2 89 90 101 6 37 8 38 9 10 38 102 curf12 ( ( 𝜑𝑦𝐵 ) → ( ( 𝑦 ( 2nd𝐾 ) 𝑦 ) ‘ ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑦 ⟩ ) ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) )
104 1 2 89 90 101 6 37 8 38 curf11 ( ( 𝜑𝑦𝐵 ) → ( ( 1st𝐾 ) ‘ 𝑦 ) = ( 𝑋 ( 1st𝐹 ) 𝑦 ) )
105 104 65 eqtrdi ( ( 𝜑𝑦𝐵 ) → ( ( 1st𝐾 ) ‘ 𝑦 ) = ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) )
106 105 fveq2d ( ( 𝜑𝑦𝐵 ) → ( ( Id ‘ 𝐸 ) ‘ ( ( 1st𝐾 ) ‘ 𝑦 ) ) = ( ( Id ‘ 𝐸 ) ‘ ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) ) )
107 100 103 106 3eqtr4d ( ( 𝜑𝑦𝐵 ) → ( ( 𝑦 ( 2nd𝐾 ) 𝑦 ) ‘ ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) = ( ( Id ‘ 𝐸 ) ‘ ( ( 1st𝐾 ) ‘ 𝑦 ) ) )
108 7 3ad2ant1 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝑋𝐴 )
109 simp21 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝑦𝐵 )
110 simp22 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝑧𝐵 )
111 eqid ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 )
112 eqid ( comp ‘ ( 𝐶 ×c 𝐷 ) ) = ( comp ‘ ( 𝐶 ×c 𝐷 ) )
113 simp23 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝑤𝐵 )
114 3 3ad2ant1 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝐶 ∈ Cat )
115 2 59 10 114 108 catidcl ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) )
116 simp3l ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) )
117 simp3r ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) )
118 30 2 6 59 9 108 109 108 110 111 25 112 108 113 115 116 115 117 xpcco2 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ ( ⟨ ⟨ 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 ) ) , ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ⟩ )
119 2 59 10 114 108 111 108 115 catlid ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) )
120 119 opeq1d ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ⟨ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) , ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ⟩ = ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ⟩ )
121 118 120 eqtrd ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ ( ⟨ ⟨ 𝑋 , 𝑦 ⟩ , ⟨ 𝑋 , 𝑧 ⟩ ⟩ ( comp ‘ ( 𝐶 ×c 𝐷 ) ) ⟨ 𝑋 , 𝑤 ⟩ ) ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 ⟩ ) = ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ⟩ )
122 121 fveq2d ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ‘ ( ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ ( ⟨ ⟨ 𝑋 , 𝑦 ⟩ , ⟨ 𝑋 , 𝑧 ⟩ ⟩ ( comp ‘ ( 𝐶 ×c 𝐷 ) ) ⟨ 𝑋 , 𝑤 ⟩ ) ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 ⟩ ) ) = ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ⟩ ) )
123 df-ov ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) = ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ⟩ )
124 122 123 eqtr4di ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ‘ ( ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ ( ⟨ ⟨ 𝑋 , 𝑦 ⟩ , ⟨ 𝑋 , 𝑧 ⟩ ⟩ ( comp ‘ ( 𝐶 ×c 𝐷 ) ) ⟨ 𝑋 , 𝑤 ⟩ ) ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 ⟩ ) ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) )
125 34 3ad2ant1 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( 1st𝐹 ) ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ( 2nd𝐹 ) )
126 108 109 opelxpd ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ⟨ 𝑋 , 𝑦 ⟩ ∈ ( 𝐴 × 𝐵 ) )
127 108 110 opelxpd ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ⟨ 𝑋 , 𝑧 ⟩ ∈ ( 𝐴 × 𝐵 ) )
128 108 113 opelxpd ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ⟨ 𝑋 , 𝑤 ⟩ ∈ ( 𝐴 × 𝐵 ) )
129 115 116 opelxpd ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 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 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ⟨ 𝑋 , 𝑦 ⟩ ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) ⟨ 𝑋 , 𝑧 ⟩ ) = ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) )
131 129 130 eleqtrrd ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 ⟩ ∈ ( ⟨ 𝑋 , 𝑦 ⟩ ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) ⟨ 𝑋 , 𝑧 ⟩ ) )
132 115 117 opelxpd ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ ∈ ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) )
133 30 2 6 59 9 108 110 108 113 51 xpchom2 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ⟨ 𝑋 , 𝑧 ⟩ ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) ⟨ 𝑋 , 𝑤 ⟩ ) = ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) )
134 132 133 eleqtrrd ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ ∈ ( ⟨ 𝑋 , 𝑧 ⟩ ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) ⟨ 𝑋 , 𝑤 ⟩ ) )
135 31 51 112 26 125 126 127 128 131 134 funcco ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ‘ ( ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ ( ⟨ ⟨ 𝑋 , 𝑦 ⟩ , ⟨ 𝑋 , 𝑧 ⟩ ⟩ ( comp ‘ ( 𝐶 ×c 𝐷 ) ) ⟨ 𝑋 , 𝑤 ⟩ ) ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 ⟩ ) ) = ( ( ( ⟨ 𝑋 , 𝑧 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ ) ( ⟨ ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) , ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ⟩ ( comp ‘ 𝐸 ) ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑤 ⟩ ) ) ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 ⟩ ) ) )
136 124 135 eqtr3d ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) = ( ( ( ⟨ 𝑋 , 𝑧 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ ) ( ⟨ ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) , ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ⟩ ( comp ‘ 𝐸 ) ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑤 ⟩ ) ) ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 ⟩ ) ) )
137 4 3ad2ant1 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝐷 ∈ Cat )
138 5 3ad2ant1 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) )
139 6 9 25 137 109 110 113 116 117 catcocl ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( ⟨ 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 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑦 ( 2nd𝐾 ) 𝑤 ) ‘ ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) )
141 1 2 114 137 138 6 108 8 109 curf11 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st𝐾 ) ‘ 𝑦 ) = ( 𝑋 ( 1st𝐹 ) 𝑦 ) )
142 141 65 eqtrdi ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st𝐾 ) ‘ 𝑦 ) = ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) )
143 1 2 114 137 138 6 108 8 110 curf11 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st𝐾 ) ‘ 𝑧 ) = ( 𝑋 ( 1st𝐹 ) 𝑧 ) )
144 143 68 eqtrdi ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st𝐾 ) ‘ 𝑧 ) = ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑧 ⟩ ) )
145 142 144 opeq12d ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ⟨ ( ( 1st𝐾 ) ‘ 𝑦 ) , ( ( 1st𝐾 ) ‘ 𝑧 ) ⟩ = ⟨ ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) , ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ⟩ )
146 1 2 114 137 138 6 108 8 113 curf11 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st𝐾 ) ‘ 𝑤 ) = ( 𝑋 ( 1st𝐹 ) 𝑤 ) )
147 df-ov ( 𝑋 ( 1st𝐹 ) 𝑤 ) = ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑤 ⟩ )
148 146 147 eqtrdi ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st𝐾 ) ‘ 𝑤 ) = ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑤 ⟩ ) )
149 145 148 oveq12d ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ⟨ ( ( 1st𝐾 ) ‘ 𝑦 ) , ( ( 1st𝐾 ) ‘ 𝑧 ) ⟩ ( comp ‘ 𝐸 ) ( ( 1st𝐾 ) ‘ 𝑤 ) ) = ( ⟨ ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) , ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ⟩ ( comp ‘ 𝐸 ) ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑤 ⟩ ) ) )
150 1 2 114 137 138 6 108 8 110 9 10 113 117 curf12 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑧 ( 2nd𝐾 ) 𝑤 ) ‘ ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑧 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ) )
151 df-ov ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑧 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ) = ( ( ⟨ 𝑋 , 𝑧 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ )
152 150 151 eqtrdi ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑧 ( 2nd𝐾 ) 𝑤 ) ‘ ) = ( ( ⟨ 𝑋 , 𝑧 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ ) )
153 1 2 114 137 138 6 108 8 109 9 10 110 116 curf12 ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑦 ( 2nd𝐾 ) 𝑧 ) ‘ 𝑔 ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) )
154 df-ov ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) 𝑔 ) = ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 ⟩ )
155 153 154 eqtrdi ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑦 ( 2nd𝐾 ) 𝑧 ) ‘ 𝑔 ) = ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 ⟩ ) )
156 149 152 155 oveq123d ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( ( 𝑧 ( 2nd𝐾 ) 𝑤 ) ‘ ) ( ⟨ ( ( 1st𝐾 ) ‘ 𝑦 ) , ( ( 1st𝐾 ) ‘ 𝑧 ) ⟩ ( comp ‘ 𝐸 ) ( ( 1st𝐾 ) ‘ 𝑤 ) ) ( ( 𝑦 ( 2nd𝐾 ) 𝑧 ) ‘ 𝑔 ) ) = ( ( ( ⟨ 𝑋 , 𝑧 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑤 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ⟩ ) ( ⟨ ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑦 ⟩ ) , ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑧 ⟩ ) ⟩ ( comp ‘ 𝐸 ) ( ( 1st𝐹 ) ‘ ⟨ 𝑋 , 𝑤 ⟩ ) ) ( ( ⟨ 𝑋 , 𝑦 ⟩ ( 2nd𝐹 ) ⟨ 𝑋 , 𝑧 ⟩ ) ‘ ⟨ ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 ⟩ ) ) )
157 136 140 156 3eqtr4d ( ( 𝜑 ∧ ( 𝑦𝐵𝑧𝐵𝑤𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑦 ( 2nd𝐾 ) 𝑤 ) ‘ ( ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) = ( ( ( 𝑧 ( 2nd𝐾 ) 𝑤 ) ‘ ) ( ⟨ ( ( 1st𝐾 ) ‘ 𝑦 ) , ( ( 1st𝐾 ) ‘ 𝑧 ) ⟩ ( comp ‘ 𝐸 ) ( ( 1st𝐾 ) ‘ 𝑤 ) ) ( ( 𝑦 ( 2nd𝐾 ) 𝑧 ) ‘ 𝑔 ) ) )
158 6 21 9 22 23 24 25 26 4 29 40 46 88 107 157 isfuncd ( 𝜑 → ( 1st𝐾 ) ( 𝐷 Func 𝐸 ) ( 2nd𝐾 ) )
159 df-br ( ( 1st𝐾 ) ( 𝐷 Func 𝐸 ) ( 2nd𝐾 ) ↔ ⟨ ( 1st𝐾 ) , ( 2nd𝐾 ) ⟩ ∈ ( 𝐷 Func 𝐸 ) )
160 158 159 sylib ( 𝜑 → ⟨ ( 1st𝐾 ) , ( 2nd𝐾 ) ⟩ ∈ ( 𝐷 Func 𝐸 ) )
161 20 160 eqeltrd ( 𝜑𝐾 ∈ ( 𝐷 Func 𝐸 ) )