Metamath Proof Explorer

Theorem evlfcl

Description: The evaluation functor is a bifunctor (a two-argument functor) with the first parameter taking values in the set of functors C --> D , and the second parameter in D . (Contributed by Mario Carneiro, 12-Jan-2017)

Ref Expression
Hypotheses evlfcl.e 
|- E = ( C evalF D )
evlfcl.q 
|- Q = ( C FuncCat D )
evlfcl.c 
|- ( ph -> C e. Cat )
evlfcl.d 
|- ( ph -> D e. Cat )
Assertion evlfcl 
|- ( ph -> E e. ( ( Q Xc. C ) Func D ) )

Proof

Step Hyp Ref Expression
1 evlfcl.e 
 |-  E = ( C evalF D )
2 evlfcl.q 
 |-  Q = ( C FuncCat D )
3 evlfcl.c 
 |-  ( ph -> C e. Cat )
4 evlfcl.d 
 |-  ( ph -> D e. Cat )
5 eqid 
 |-  ( Base ` C ) = ( Base ` C )
6 eqid 
 |-  ( Hom ` C ) = ( Hom ` C )
7 eqid 
 |-  ( comp ` D ) = ( comp ` D )
8 eqid 
 |-  ( C Nat D ) = ( C Nat D )
9 1 3 4 5 6 7 8 evlfval 
 |-  ( ph -> E = <. ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. )
10 ovex 
 |-  ( C Func D ) e. _V
11 fvex 
 |-  ( Base ` C ) e. _V
12 10 11 mpoex 
 |-  ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) e. _V
13 10 11 xpex 
 |-  ( ( C Func D ) X. ( Base ` C ) ) e. _V
14 13 13 mpoex 
 |-  ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) e. _V
15 12 14 opelvv 
 |-  <. ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. e. ( _V X. _V )
16 9 15 eqeltrdi 
 |-  ( ph -> E e. ( _V X. _V ) )
17 1st2nd2 
 |-  ( E e. ( _V X. _V ) -> E = <. ( 1st ` E ) , ( 2nd ` E ) >. )
18 16 17 syl 
 |-  ( ph -> E = <. ( 1st ` E ) , ( 2nd ` E ) >. )
19 eqid 
 |-  ( Q Xc. C ) = ( Q Xc. C )
20 2 fucbas 
 |-  ( C Func D ) = ( Base ` Q )
21 19 20 5 xpcbas 
 |-  ( ( C Func D ) X. ( Base ` C ) ) = ( Base ` ( Q Xc. C ) )
22 eqid 
 |-  ( Base ` D ) = ( Base ` D )
23 eqid 
 |-  ( Hom ` ( Q Xc. C ) ) = ( Hom ` ( Q Xc. C ) )
24 eqid 
 |-  ( Hom ` D ) = ( Hom ` D )
25 eqid 
 |-  ( Id ` ( Q Xc. C ) ) = ( Id ` ( Q Xc. C ) )
26 eqid 
 |-  ( Id ` D ) = ( Id ` D )
27 eqid 
 |-  ( comp ` ( Q Xc. C ) ) = ( comp ` ( Q Xc. C ) )
28 2 3 4 fuccat 
 |-  ( ph -> Q e. Cat )
29 19 28 3 xpccat 
 |-  ( ph -> ( Q Xc. C ) e. Cat )
30 relfunc 
 |-  Rel ( C Func D )
31 simpr 
 |-  ( ( ph /\ f e. ( C Func D ) ) -> f e. ( C Func D ) )
32 1st2ndbr 
 |-  ( ( Rel ( C Func D ) /\ f e. ( C Func D ) ) -> ( 1st ` f ) ( C Func D ) ( 2nd ` f ) )
33 30 31 32 sylancr 
 |-  ( ( ph /\ f e. ( C Func D ) ) -> ( 1st ` f ) ( C Func D ) ( 2nd ` f ) )
34 5 22 33 funcf1 
 |-  ( ( ph /\ f e. ( C Func D ) ) -> ( 1st ` f ) : ( Base ` C ) --> ( Base ` D ) )
35 34 ffvelrnda 
 |-  ( ( ( ph /\ f e. ( C Func D ) ) /\ x e. ( Base ` C ) ) -> ( ( 1st ` f ) ` x ) e. ( Base ` D ) )
36 35 ralrimiva 
 |-  ( ( ph /\ f e. ( C Func D ) ) -> A. x e. ( Base ` C ) ( ( 1st ` f ) ` x ) e. ( Base ` D ) )
37 36 ralrimiva 
 |-  ( ph -> A. f e. ( C Func D ) A. x e. ( Base ` C ) ( ( 1st ` f ) ` x ) e. ( Base ` D ) )
38 eqid 
 |-  ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) = ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) )
39 38 fmpo 
 |-  ( A. f e. ( C Func D ) A. x e. ( Base ` C ) ( ( 1st ` f ) ` x ) e. ( Base ` D ) <-> ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) : ( ( C Func D ) X. ( Base ` C ) ) --> ( Base ` D ) )
40 37 39 sylib 
 |-  ( ph -> ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) : ( ( C Func D ) X. ( Base ` C ) ) --> ( Base ` D ) )
41 12 14 op1std 
 |-  ( E = <. ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. -> ( 1st ` E ) = ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) )
42 9 41 syl 
 |-  ( ph -> ( 1st ` E ) = ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) )
43 42 feq1d 
 |-  ( ph -> ( ( 1st ` E ) : ( ( C Func D ) X. ( Base ` C ) ) --> ( Base ` D ) <-> ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) : ( ( C Func D ) X. ( Base ` C ) ) --> ( Base ` D ) ) )
44 40 43 mpbird 
 |-  ( ph -> ( 1st ` E ) : ( ( C Func D ) X. ( Base ` C ) ) --> ( Base ` D ) )
45 eqid 
 |-  ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) = ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) )
46 ovex 
 |-  ( m ( C Nat D ) n ) e. _V
47 ovex 
 |-  ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) e. _V
48 46 47 mpoex 
 |-  ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) e. _V
49 48 csbex 
 |-  [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) e. _V
50 49 csbex 
 |-  [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) e. _V
51 45 50 fnmpoi 
 |-  ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) Fn ( ( ( C Func D ) X. ( Base ` C ) ) X. ( ( C Func D ) X. ( Base ` C ) ) )
52 12 14 op2ndd 
 |-  ( E = <. ( f e. ( C Func D ) , x e. ( Base ` C ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. -> ( 2nd ` E ) = ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) )
53 9 52 syl 
 |-  ( ph -> ( 2nd ` E ) = ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) )
54 53 fneq1d 
 |-  ( ph -> ( ( 2nd ` E ) Fn ( ( ( C Func D ) X. ( Base ` C ) ) X. ( ( C Func D ) X. ( Base ` C ) ) ) <-> ( x e. ( ( C Func D ) X. ( Base ` C ) ) , y e. ( ( C Func D ) X. ( Base ` C ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. ( comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) Fn ( ( ( C Func D ) X. ( Base ` C ) ) X. ( ( C Func D ) X. ( Base ` C ) ) ) ) )
55 51 54 mpbiri 
 |-  ( ph -> ( 2nd ` E ) Fn ( ( ( C Func D ) X. ( Base ` C ) ) X. ( ( C Func D ) X. ( Base ` C ) ) ) )
56 4 ad2antrr 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> D e. Cat )
57 56 adantr 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> D e. Cat )
58 simplrl 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> f e. ( C Func D ) )
59 30 58 32 sylancr 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( 1st ` f ) ( C Func D ) ( 2nd ` f ) )
60 5 22 59 funcf1 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( 1st ` f ) : ( Base ` C ) --> ( Base ` D ) )
61 60 adantr 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( 1st ` f ) : ( Base ` C ) --> ( Base ` D ) )
62 simplrr 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> u e. ( Base ` C ) )
63 62 adantr 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> u e. ( Base ` C ) )
64 61 63 ffvelrnd 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( ( 1st ` f ) ` u ) e. ( Base ` D ) )
65 simplrr 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> v e. ( Base ` C ) )
66 61 65 ffvelrnd 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( ( 1st ` f ) ` v ) e. ( Base ` D ) )
67 simprl 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> g e. ( C Func D ) )
68 1st2ndbr 
 |-  ( ( Rel ( C Func D ) /\ g e. ( C Func D ) ) -> ( 1st ` g ) ( C Func D ) ( 2nd ` g ) )
69 30 67 68 sylancr 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( 1st ` g ) ( C Func D ) ( 2nd ` g ) )
70 5 22 69 funcf1 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( 1st ` g ) : ( Base ` C ) --> ( Base ` D ) )
71 70 adantr 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( 1st ` g ) : ( Base ` C ) --> ( Base ` D ) )
72 71 65 ffvelrnd 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( ( 1st ` g ) ` v ) e. ( Base ` D ) )
73 simprr 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> v e. ( Base ` C ) )
74 5 6 24 59 62 73 funcf2 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( u ( 2nd ` f ) v ) : ( u ( Hom ` C ) v ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` f ) ` v ) ) )
75 74 adantr 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( u ( 2nd ` f ) v ) : ( u ( Hom ` C ) v ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` f ) ` v ) ) )
76 simprr 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> h e. ( u ( Hom ` C ) v ) )
77 75 76 ffvelrnd 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( ( u ( 2nd ` f ) v ) ` h ) e. ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` f ) ` v ) ) )
78 simprl 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> a e. ( f ( C Nat D ) g ) )
79 8 78 nat1st2nd 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> a e. ( <. ( 1st ` f ) , ( 2nd ` f ) >. ( C Nat D ) <. ( 1st ` g ) , ( 2nd ` g ) >. ) )
80 8 79 5 24 65 natcl 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( a ` v ) e. ( ( ( 1st ` f ) ` v ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
81 22 24 7 57 64 66 72 77 80 catcocl 
 |-  ( ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) /\ ( a e. ( f ( C Nat D ) g ) /\ h e. ( u ( Hom ` C ) v ) ) ) -> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. ( comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) e. ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
82 81 ralrimivva 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> A. a e. ( f ( C Nat D ) g ) A. h e. ( u ( Hom ` C ) v ) ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. ( comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) e. ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
83 eqid 
 |-  ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. ( comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) ) = ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. ( comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) )
84 83 fmpo 
 |-  ( A. a e. ( f ( C Nat D ) g ) A. h e. ( u ( Hom ` C ) v ) ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. ( comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) e. ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) <-> ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. ( comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
85 82 84 sylib 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. ( comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
86 3 ad2antrr 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> C e. Cat )
87 eqid 
 |-  ( <. f , u >. ( 2nd ` E ) <. g , v >. ) = ( <. f , u >. ( 2nd ` E ) <. g , v >. )
88 1 86 56 5 6 7 8 58 67 62 73 87 evlf2 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( <. f , u >. ( 2nd ` E ) <. g , v >. ) = ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. ( comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) ) )
89 88 feq1d 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) <-> ( a e. ( f ( C Nat D ) g ) , h e. ( u ( Hom ` C ) v ) |-> ( ( a ` v ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` v ) >. ( comp ` D ) ( ( 1st ` g ) ` v ) ) ( ( u ( 2nd ` f ) v ) ` h ) ) ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) ) )
90 85 89 mpbird 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
91 2 8 fuchom 
 |-  ( C Nat D ) = ( Hom ` Q )
92 19 20 5 91 6 58 62 67 73 23 xpchom2 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( <. f , u >. ( Hom ` ( Q Xc. C ) ) <. g , v >. ) = ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) )
93 1 86 56 5 58 62 evlf1 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( f ( 1st ` E ) u ) = ( ( 1st ` f ) ` u ) )
94 1 86 56 5 67 73 evlf1 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( g ( 1st ` E ) v ) = ( ( 1st ` g ) ` v ) )
95 93 94 oveq12d 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) = ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) )
96 92 95 feq23d 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) <-> ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( ( f ( C Nat D ) g ) X. ( u ( Hom ` C ) v ) ) --> ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` g ) ` v ) ) ) )
97 90 96 mpbird 
 |-  ( ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) /\ ( g e. ( C Func D ) /\ v e. ( Base ` C ) ) ) -> ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
98 97 ralrimivva 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> A. g e. ( C Func D ) A. v e. ( Base ` C ) ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
99 98 ralrimivva 
 |-  ( ph -> A. f e. ( C Func D ) A. u e. ( Base ` C ) A. g e. ( C Func D ) A. v e. ( Base ` C ) ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
100 oveq2 
 |-  ( y = <. g , v >. -> ( x ( 2nd ` E ) y ) = ( x ( 2nd ` E ) <. g , v >. ) )
101 oveq2 
 |-  ( y = <. g , v >. -> ( x ( Hom ` ( Q Xc. C ) ) y ) = ( x ( Hom ` ( Q Xc. C ) ) <. g , v >. ) )
102 fveq2 
 |-  ( y = <. g , v >. -> ( ( 1st ` E ) ` y ) = ( ( 1st ` E ) ` <. g , v >. ) )
103 df-ov 
 |-  ( g ( 1st ` E ) v ) = ( ( 1st ` E ) ` <. g , v >. )
104 102 103 eqtr4di 
 |-  ( y = <. g , v >. -> ( ( 1st ` E ) ` y ) = ( g ( 1st ` E ) v ) )
105 104 oveq2d 
 |-  ( y = <. g , v >. -> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) = ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
106 100 101 105 feq123d 
 |-  ( y = <. g , v >. -> ( ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q Xc. C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) <-> ( x ( 2nd ` E ) <. g , v >. ) : ( x ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) ) )
107 106 ralxp 
 |-  ( A. y e. ( ( C Func D ) X. ( Base ` C ) ) ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q Xc. C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) <-> A. g e. ( C Func D ) A. v e. ( Base ` C ) ( x ( 2nd ` E ) <. g , v >. ) : ( x ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
108 oveq1 
 |-  ( x = <. f , u >. -> ( x ( 2nd ` E ) <. g , v >. ) = ( <. f , u >. ( 2nd ` E ) <. g , v >. ) )
109 oveq1 
 |-  ( x = <. f , u >. -> ( x ( Hom ` ( Q Xc. C ) ) <. g , v >. ) = ( <. f , u >. ( Hom ` ( Q Xc. C ) ) <. g , v >. ) )
110 fveq2 
 |-  ( x = <. f , u >. -> ( ( 1st ` E ) ` x ) = ( ( 1st ` E ) ` <. f , u >. ) )
111 df-ov 
 |-  ( f ( 1st ` E ) u ) = ( ( 1st ` E ) ` <. f , u >. )
112 110 111 eqtr4di 
 |-  ( x = <. f , u >. -> ( ( 1st ` E ) ` x ) = ( f ( 1st ` E ) u ) )
113 112 oveq1d 
 |-  ( x = <. f , u >. -> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) = ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
114 108 109 113 feq123d 
 |-  ( x = <. f , u >. -> ( ( x ( 2nd ` E ) <. g , v >. ) : ( x ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) <-> ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) ) )
115 114 2ralbidv 
 |-  ( x = <. f , u >. -> ( A. g e. ( C Func D ) A. v e. ( Base ` C ) ( x ( 2nd ` E ) <. g , v >. ) : ( x ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) <-> A. g e. ( C Func D ) A. v e. ( Base ` C ) ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) ) )
116 107 115 syl5bb 
 |-  ( x = <. f , u >. -> ( A. y e. ( ( C Func D ) X. ( Base ` C ) ) ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q Xc. C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) <-> A. g e. ( C Func D ) A. v e. ( Base ` C ) ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) ) )
117 116 ralxp 
 |-  ( A. x e. ( ( C Func D ) X. ( Base ` C ) ) A. y e. ( ( C Func D ) X. ( Base ` C ) ) ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q Xc. C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) <-> A. f e. ( C Func D ) A. u e. ( Base ` C ) A. g e. ( C Func D ) A. v e. ( Base ` C ) ( <. f , u >. ( 2nd ` E ) <. g , v >. ) : ( <. f , u >. ( Hom ` ( Q Xc. C ) ) <. g , v >. ) --> ( ( f ( 1st ` E ) u ) ( Hom ` D ) ( g ( 1st ` E ) v ) ) )
118 99 117 sylibr 
 |-  ( ph -> A. x e. ( ( C Func D ) X. ( Base ` C ) ) A. y e. ( ( C Func D ) X. ( Base ` C ) ) ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q Xc. C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) )
119 118 r19.21bi 
 |-  ( ( ph /\ x e. ( ( C Func D ) X. ( Base ` C ) ) ) -> A. y e. ( ( C Func D ) X. ( Base ` C ) ) ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q Xc. C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) )
120 119 r19.21bi 
 |-  ( ( ( ph /\ x e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) ) -> ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q Xc. C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) )
121 120 anasss 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) ) ) -> ( x ( 2nd ` E ) y ) : ( x ( Hom ` ( Q Xc. C ) ) y ) --> ( ( ( 1st ` E ) ` x ) ( Hom ` D ) ( ( 1st ` E ) ` y ) ) )
122 28 adantr 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> Q e. Cat )
123 3 adantr 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> C e. Cat )
124 eqid 
 |-  ( Id ` Q ) = ( Id ` Q )
125 eqid 
 |-  ( Id ` C ) = ( Id ` C )
126 simprl 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> f e. ( C Func D ) )
127 simprr 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> u e. ( Base ` C ) )
128 19 122 123 20 5 124 125 25 126 127 xpcid 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` ( Q Xc. C ) ) ` <. f , u >. ) = <. ( ( Id ` Q ) ` f ) , ( ( Id ` C ) ` u ) >. )
129 128 fveq2d 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q Xc. C ) ) ` <. f , u >. ) ) = ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` <. ( ( Id ` Q ) ` f ) , ( ( Id ` C ) ` u ) >. ) )
130 df-ov 
 |-  ( ( ( Id ` Q ) ` f ) ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ( ( Id ` C ) ` u ) ) = ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` <. ( ( Id ` Q ) ` f ) , ( ( Id ` C ) ` u ) >. )
131 129 130 eqtr4di 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q Xc. C ) ) ` <. f , u >. ) ) = ( ( ( Id ` Q ) ` f ) ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ( ( Id ` C ) ` u ) ) )
132 4 adantr 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> D e. Cat )
133 eqid 
 |-  ( <. f , u >. ( 2nd ` E ) <. f , u >. ) = ( <. f , u >. ( 2nd ` E ) <. f , u >. )
134 20 91 124 122 126 catidcl 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` Q ) ` f ) e. ( f ( C Nat D ) f ) )
135 5 6 125 123 127 catidcl 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` C ) ` u ) e. ( u ( Hom ` C ) u ) )
136 1 123 132 5 6 7 8 126 126 127 127 133 134 135 evlf2val 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( Id ` Q ) ` f ) ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ( ( Id ` C ) ` u ) ) = ( ( ( ( Id ` Q ) ` f ) ` u ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` u ) >. ( comp ` D ) ( ( 1st ` f ) ` u ) ) ( ( u ( 2nd ` f ) u ) ` ( ( Id ` C ) ` u ) ) ) )
137 30 126 32 sylancr 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( 1st ` f ) ( C Func D ) ( 2nd ` f ) )
138 5 22 137 funcf1 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( 1st ` f ) : ( Base ` C ) --> ( Base ` D ) )
139 138 127 ffvelrnd 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( 1st ` f ) ` u ) e. ( Base ` D ) )
140 22 24 26 132 139 catidcl 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) e. ( ( ( 1st ` f ) ` u ) ( Hom ` D ) ( ( 1st ` f ) ` u ) ) )
141 22 24 26 132 139 7 139 140 catlid 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` u ) >. ( comp ` D ) ( ( 1st ` f ) ` u ) ) ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
142 2 124 26 126 fucid 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` Q ) ` f ) = ( ( Id ` D ) o. ( 1st ` f ) ) )
143 142 fveq1d 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( Id ` Q ) ` f ) ` u ) = ( ( ( Id ` D ) o. ( 1st ` f ) ) ` u ) )
144 fvco3 
 |-  ( ( ( 1st ` f ) : ( Base ` C ) --> ( Base ` D ) /\ u e. ( Base ` C ) ) -> ( ( ( Id ` D ) o. ( 1st ` f ) ) ` u ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
145 138 127 144 syl2anc 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( Id ` D ) o. ( 1st ` f ) ) ` u ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
146 143 145 eqtrd 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( Id ` Q ) ` f ) ` u ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
147 5 125 26 137 127 funcid 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( u ( 2nd ` f ) u ) ` ( ( Id ` C ) ` u ) ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
148 146 147 oveq12d 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( ( Id ` Q ) ` f ) ` u ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` u ) >. ( comp ` D ) ( ( 1st ` f ) ` u ) ) ( ( u ( 2nd ` f ) u ) ` ( ( Id ` C ) ` u ) ) ) = ( ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` u ) >. ( comp ` D ) ( ( 1st ` f ) ` u ) ) ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) ) )
149 1 123 132 5 126 127 evlf1 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( f ( 1st ` E ) u ) = ( ( 1st ` f ) ` u ) )
150 149 fveq2d 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) = ( ( Id ` D ) ` ( ( 1st ` f ) ` u ) ) )
151 141 148 150 3eqtr4d 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( ( ( Id ` Q ) ` f ) ` u ) ( <. ( ( 1st ` f ) ` u ) , ( ( 1st ` f ) ` u ) >. ( comp ` D ) ( ( 1st ` f ) ` u ) ) ( ( u ( 2nd ` f ) u ) ` ( ( Id ` C ) ` u ) ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) )
152 131 136 151 3eqtrd 
 |-  ( ( ph /\ ( f e. ( C Func D ) /\ u e. ( Base ` C ) ) ) -> ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q Xc. C ) ) ` <. f , u >. ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) )
153 152 ralrimivva 
 |-  ( ph -> A. f e. ( C Func D ) A. u e. ( Base ` C ) ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q Xc. C ) ) ` <. f , u >. ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) )
154 id 
 |-  ( x = <. f , u >. -> x = <. f , u >. )
155 154 154 oveq12d 
 |-  ( x = <. f , u >. -> ( x ( 2nd ` E ) x ) = ( <. f , u >. ( 2nd ` E ) <. f , u >. ) )
156 fveq2 
 |-  ( x = <. f , u >. -> ( ( Id ` ( Q Xc. C ) ) ` x ) = ( ( Id ` ( Q Xc. C ) ) ` <. f , u >. ) )
157 155 156 fveq12d 
 |-  ( x = <. f , u >. -> ( ( x ( 2nd ` E ) x ) ` ( ( Id ` ( Q Xc. C ) ) ` x ) ) = ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q Xc. C ) ) ` <. f , u >. ) ) )
158 112 fveq2d 
 |-  ( x = <. f , u >. -> ( ( Id ` D ) ` ( ( 1st ` E ) ` x ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) )
159 157 158 eqeq12d 
 |-  ( x = <. f , u >. -> ( ( ( x ( 2nd ` E ) x ) ` ( ( Id ` ( Q Xc. C ) ) ` x ) ) = ( ( Id ` D ) ` ( ( 1st ` E ) ` x ) ) <-> ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q Xc. C ) ) ` <. f , u >. ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) ) )
160 159 ralxp 
 |-  ( A. x e. ( ( C Func D ) X. ( Base ` C ) ) ( ( x ( 2nd ` E ) x ) ` ( ( Id ` ( Q Xc. C ) ) ` x ) ) = ( ( Id ` D ) ` ( ( 1st ` E ) ` x ) ) <-> A. f e. ( C Func D ) A. u e. ( Base ` C ) ( ( <. f , u >. ( 2nd ` E ) <. f , u >. ) ` ( ( Id ` ( Q Xc. C ) ) ` <. f , u >. ) ) = ( ( Id ` D ) ` ( f ( 1st ` E ) u ) ) )
161 153 160 sylibr 
 |-  ( ph -> A. x e. ( ( C Func D ) X. ( Base ` C ) ) ( ( x ( 2nd ` E ) x ) ` ( ( Id ` ( Q Xc. C ) ) ` x ) ) = ( ( Id ` D ) ` ( ( 1st ` E ) ` x ) ) )
162 161 r19.21bi 
 |-  ( ( ph /\ x e. ( ( C Func D ) X. ( Base ` C ) ) ) -> ( ( x ( 2nd ` E ) x ) ` ( ( Id ` ( Q Xc. C ) ) ` x ) ) = ( ( Id ` D ) ` ( ( 1st ` E ) ` x ) ) )
163 3 3ad2ant1 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> C e. Cat )
164 4 3ad2ant1 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> D e. Cat )
165 simp21 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> x e. ( ( C Func D ) X. ( Base ` C ) ) )
166 1st2nd2 
 |-  ( x e. ( ( C Func D ) X. ( Base ` C ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
167 165 166 syl 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
168 167 165 eqeltrrd 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( C Func D ) X. ( Base ` C ) ) )
169 opelxp 
 |-  ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( C Func D ) X. ( Base ` C ) ) <-> ( ( 1st ` x ) e. ( C Func D ) /\ ( 2nd ` x ) e. ( Base ` C ) ) )
170 168 169 sylib 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( 1st ` x ) e. ( C Func D ) /\ ( 2nd ` x ) e. ( Base ` C ) ) )
171 simp22 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> y e. ( ( C Func D ) X. ( Base ` C ) ) )
172 1st2nd2 
 |-  ( y e. ( ( C Func D ) X. ( Base ` C ) ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
173 171 172 syl 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
174 173 171 eqeltrrd 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. ( ( C Func D ) X. ( Base ` C ) ) )
175 opelxp 
 |-  ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. ( ( C Func D ) X. ( Base ` C ) ) <-> ( ( 1st ` y ) e. ( C Func D ) /\ ( 2nd ` y ) e. ( Base ` C ) ) )
176 174 175 sylib 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( 1st ` y ) e. ( C Func D ) /\ ( 2nd ` y ) e. ( Base ` C ) ) )
177 simp23 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> z e. ( ( C Func D ) X. ( Base ` C ) ) )
178 1st2nd2 
 |-  ( z e. ( ( C Func D ) X. ( Base ` C ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
179 177 178 syl 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
180 179 177 eqeltrrd 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. ( ( C Func D ) X. ( Base ` C ) ) )
181 opelxp 
 |-  ( <. ( 1st ` z ) , ( 2nd ` z ) >. e. ( ( C Func D ) X. ( Base ` C ) ) <-> ( ( 1st ` z ) e. ( C Func D ) /\ ( 2nd ` z ) e. ( Base ` C ) ) )
182 180 181 sylib 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( 1st ` z ) e. ( C Func D ) /\ ( 2nd ` z ) e. ( Base ` C ) ) )
183 simp3l 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> f e. ( x ( Hom ` ( Q Xc. C ) ) y ) )
184 19 21 91 6 23 165 171 xpchom 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( x ( Hom ` ( Q Xc. C ) ) y ) = ( ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
185 183 184 eleqtrd 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> f e. ( ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
186 1st2nd2 
 |-  ( f e. ( ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) -> f = <. ( 1st ` f ) , ( 2nd ` f ) >. )
187 185 186 syl 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> f = <. ( 1st ` f ) , ( 2nd ` f ) >. )
188 187 185 eqeltrrd 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> <. ( 1st ` f ) , ( 2nd ` f ) >. e. ( ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
189 opelxp 
 |-  ( <. ( 1st ` f ) , ( 2nd ` f ) >. e. ( ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) <-> ( ( 1st ` f ) e. ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) /\ ( 2nd ` f ) e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
190 188 189 sylib 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( 1st ` f ) e. ( ( 1st ` x ) ( C Nat D ) ( 1st ` y ) ) /\ ( 2nd ` f ) e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
191 simp3r 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> g e. ( y ( Hom ` ( Q Xc. C ) ) z ) )
192 19 21 91 6 23 171 177 xpchom 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( y ( Hom ` ( Q Xc. C ) ) z ) = ( ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
193 191 192 eleqtrd 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> g e. ( ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
194 1st2nd2 
 |-  ( g e. ( ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) -> g = <. ( 1st ` g ) , ( 2nd ` g ) >. )
195 193 194 syl 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> g = <. ( 1st ` g ) , ( 2nd ` g ) >. )
196 195 193 eqeltrrd 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> <. ( 1st ` g ) , ( 2nd ` g ) >. e. ( ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
197 opelxp 
 |-  ( <. ( 1st ` g ) , ( 2nd ` g ) >. e. ( ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) <-> ( ( 1st ` g ) e. ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) /\ ( 2nd ` g ) e. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
198 196 197 sylib 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( 1st ` g ) e. ( ( 1st ` y ) ( C Nat D ) ( 1st ` z ) ) /\ ( 2nd ` g ) e. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
199 1 2 163 164 8 170 176 182 190 198 evlfcllem 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` ( <. ( 1st ` g ) , ( 2nd ` g ) >. ( <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. ( comp ` ( Q Xc. C ) ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) <. ( 1st ` f ) , ( 2nd ` f ) >. ) ) = ( ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. ) ( <. ( ( 1st ` E ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) , ( ( 1st ` E ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) >. ( comp ` D ) ( ( 1st ` E ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) ) ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ` <. ( 1st ` f ) , ( 2nd ` f ) >. ) ) )
200 167 179 oveq12d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( x ( 2nd ` E ) z ) = ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
201 167 173 opeq12d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> <. x , y >. = <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. )
202 201 179 oveq12d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( <. x , y >. ( comp ` ( Q Xc. C ) ) z ) = ( <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. ( comp ` ( Q Xc. C ) ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
203 202 195 187 oveq123d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( g ( <. x , y >. ( comp ` ( Q Xc. C ) ) z ) f ) = ( <. ( 1st ` g ) , ( 2nd ` g ) >. ( <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. ( comp ` ( Q Xc. C ) ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) <. ( 1st ` f ) , ( 2nd ` f ) >. ) )
204 200 203 fveq12d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( x ( 2nd ` E ) z ) ` ( g ( <. x , y >. ( comp ` ( Q Xc. C ) ) z ) f ) ) = ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` ( <. ( 1st ` g ) , ( 2nd ` g ) >. ( <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. ( comp ` ( Q Xc. C ) ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) <. ( 1st ` f ) , ( 2nd ` f ) >. ) ) )
205 167 fveq2d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( 1st ` E ) ` x ) = ( ( 1st ` E ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
206 173 fveq2d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( 1st ` E ) ` y ) = ( ( 1st ` E ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) )
207 205 206 opeq12d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> <. ( ( 1st ` E ) ` x ) , ( ( 1st ` E ) ` y ) >. = <. ( ( 1st ` E ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) , ( ( 1st ` E ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) >. )
208 179 fveq2d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( 1st ` E ) ` z ) = ( ( 1st ` E ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
209 207 208 oveq12d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( <. ( ( 1st ` E ) ` x ) , ( ( 1st ` E ) ` y ) >. ( comp ` D ) ( ( 1st ` E ) ` z ) ) = ( <. ( ( 1st ` E ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) , ( ( 1st ` E ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) >. ( comp ` D ) ( ( 1st ` E ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) ) )
210 173 179 oveq12d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( y ( 2nd ` E ) z ) = ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
211 210 195 fveq12d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( y ( 2nd ` E ) z ) ` g ) = ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. ) )
212 167 173 oveq12d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( x ( 2nd ` E ) y ) = ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) )
213 212 187 fveq12d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( x ( 2nd ` E ) y ) ` f ) = ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ` <. ( 1st ` f ) , ( 2nd ` f ) >. ) )
214 209 211 213 oveq123d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( ( y ( 2nd ` E ) z ) ` g ) ( <. ( ( 1st ` E ) ` x ) , ( ( 1st ` E ) ` y ) >. ( comp ` D ) ( ( 1st ` E ) ` z ) ) ( ( x ( 2nd ` E ) y ) ` f ) ) = ( ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` E ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. ) ( <. ( ( 1st ` E ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) , ( ( 1st ` E ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) >. ( comp ` D ) ( ( 1st ` E ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) ) ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` E ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ` <. ( 1st ` f ) , ( 2nd ` f ) >. ) ) )
215 199 204 214 3eqtr4d 
 |-  ( ( ph /\ ( x e. ( ( C Func D ) X. ( Base ` C ) ) /\ y e. ( ( C Func D ) X. ( Base ` C ) ) /\ z e. ( ( C Func D ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( Q Xc. C ) ) y ) /\ g e. ( y ( Hom ` ( Q Xc. C ) ) z ) ) ) -> ( ( x ( 2nd ` E ) z ) ` ( g ( <. x , y >. ( comp ` ( Q Xc. C ) ) z ) f ) ) = ( ( ( y ( 2nd ` E ) z ) ` g ) ( <. ( ( 1st ` E ) ` x ) , ( ( 1st ` E ) ` y ) >. ( comp ` D ) ( ( 1st ` E ) ` z ) ) ( ( x ( 2nd ` E ) y ) ` f ) ) )
216 21 22 23 24 25 26 27 7 29 4 44 55 121 162 215 isfuncd 
 |-  ( ph -> ( 1st ` E ) ( ( Q Xc. C ) Func D ) ( 2nd ` E ) )
217 df-br 
 |-  ( ( 1st ` E ) ( ( Q Xc. C ) Func D ) ( 2nd ` E ) <-> <. ( 1st ` E ) , ( 2nd ` E ) >. e. ( ( Q Xc. C ) Func D ) )
218 216 217 sylib 
 |-  ( ph -> <. ( 1st ` E ) , ( 2nd ` E ) >. e. ( ( Q Xc. C ) Func D ) )
219 18 218 eqeltrd 
 |-  ( ph -> E e. ( ( Q Xc. C ) Func D ) )