Step |
Hyp |
Ref |
Expression |
1 |
|
prf1st.p |
|- P = ( F pairF G ) |
2 |
|
prf1st.c |
|- ( ph -> F e. ( C Func D ) ) |
3 |
|
prf1st.d |
|- ( ph -> G e. ( C Func E ) ) |
4 |
|
eqid |
|- ( D Xc. E ) = ( D Xc. E ) |
5 |
|
eqid |
|- ( Base ` D ) = ( Base ` D ) |
6 |
|
eqid |
|- ( Base ` E ) = ( Base ` E ) |
7 |
4 5 6
|
xpcbas |
|- ( ( Base ` D ) X. ( Base ` E ) ) = ( Base ` ( D Xc. E ) ) |
8 |
|
eqid |
|- ( Hom ` ( D Xc. E ) ) = ( Hom ` ( D Xc. E ) ) |
9 |
|
funcrcl |
|- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) ) |
10 |
2 9
|
syl |
|- ( ph -> ( C e. Cat /\ D e. Cat ) ) |
11 |
10
|
simprd |
|- ( ph -> D e. Cat ) |
12 |
11
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> D e. Cat ) |
13 |
|
funcrcl |
|- ( G e. ( C Func E ) -> ( C e. Cat /\ E e. Cat ) ) |
14 |
3 13
|
syl |
|- ( ph -> ( C e. Cat /\ E e. Cat ) ) |
15 |
14
|
simprd |
|- ( ph -> E e. Cat ) |
16 |
15
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> E e. Cat ) |
17 |
|
eqid |
|- ( D 1stF E ) = ( D 1stF E ) |
18 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
19 |
|
relfunc |
|- Rel ( C Func D ) |
20 |
|
1st2ndbr |
|- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) ) |
21 |
19 2 20
|
sylancr |
|- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) ) |
22 |
18 5 21
|
funcf1 |
|- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) ) |
23 |
22
|
ffvelrnda |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) ) |
24 |
|
relfunc |
|- Rel ( C Func E ) |
25 |
|
1st2ndbr |
|- ( ( Rel ( C Func E ) /\ G e. ( C Func E ) ) -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) ) |
26 |
24 3 25
|
sylancr |
|- ( ph -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) ) |
27 |
18 6 26
|
funcf1 |
|- ( ph -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` E ) ) |
28 |
27
|
ffvelrnda |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` E ) ) |
29 |
23 28
|
opelxpd |
|- ( ( ph /\ x e. ( Base ` C ) ) -> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. e. ( ( Base ` D ) X. ( Base ` E ) ) ) |
30 |
4 7 8 12 16 17 29
|
1stf1 |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( D 1stF E ) ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) = ( 1st ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) ) |
31 |
|
fvex |
|- ( ( 1st ` F ) ` x ) e. _V |
32 |
|
fvex |
|- ( ( 1st ` G ) ` x ) e. _V |
33 |
31 32
|
op1st |
|- ( 1st ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) = ( ( 1st ` F ) ` x ) |
34 |
30 33
|
eqtrdi |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( D 1stF E ) ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) = ( ( 1st ` F ) ` x ) ) |
35 |
34
|
mpteq2dva |
|- ( ph -> ( x e. ( Base ` C ) |-> ( ( 1st ` ( D 1stF E ) ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) ) = ( x e. ( Base ` C ) |-> ( ( 1st ` F ) ` x ) ) ) |
36 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
37 |
1 18 36 2 3
|
prfval |
|- ( ph -> P = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. ) |
38 |
|
fvex |
|- ( Base ` C ) e. _V |
39 |
38
|
mptex |
|- ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) e. _V |
40 |
38 38
|
mpoex |
|- ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) e. _V |
41 |
39 40
|
op1std |
|- ( P = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. -> ( 1st ` P ) = ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) ) |
42 |
37 41
|
syl |
|- ( ph -> ( 1st ` P ) = ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) ) |
43 |
|
relfunc |
|- Rel ( ( D Xc. E ) Func D ) |
44 |
4 11 15 17
|
1stfcl |
|- ( ph -> ( D 1stF E ) e. ( ( D Xc. E ) Func D ) ) |
45 |
|
1st2ndbr |
|- ( ( Rel ( ( D Xc. E ) Func D ) /\ ( D 1stF E ) e. ( ( D Xc. E ) Func D ) ) -> ( 1st ` ( D 1stF E ) ) ( ( D Xc. E ) Func D ) ( 2nd ` ( D 1stF E ) ) ) |
46 |
43 44 45
|
sylancr |
|- ( ph -> ( 1st ` ( D 1stF E ) ) ( ( D Xc. E ) Func D ) ( 2nd ` ( D 1stF E ) ) ) |
47 |
7 5 46
|
funcf1 |
|- ( ph -> ( 1st ` ( D 1stF E ) ) : ( ( Base ` D ) X. ( Base ` E ) ) --> ( Base ` D ) ) |
48 |
47
|
feqmptd |
|- ( ph -> ( 1st ` ( D 1stF E ) ) = ( u e. ( ( Base ` D ) X. ( Base ` E ) ) |-> ( ( 1st ` ( D 1stF E ) ) ` u ) ) ) |
49 |
|
fveq2 |
|- ( u = <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. -> ( ( 1st ` ( D 1stF E ) ) ` u ) = ( ( 1st ` ( D 1stF E ) ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) ) |
50 |
29 42 48 49
|
fmptco |
|- ( ph -> ( ( 1st ` ( D 1stF E ) ) o. ( 1st ` P ) ) = ( x e. ( Base ` C ) |-> ( ( 1st ` ( D 1stF E ) ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) ) ) |
51 |
22
|
feqmptd |
|- ( ph -> ( 1st ` F ) = ( x e. ( Base ` C ) |-> ( ( 1st ` F ) ` x ) ) ) |
52 |
35 50 51
|
3eqtr4d |
|- ( ph -> ( ( 1st ` ( D 1stF E ) ) o. ( 1st ` P ) ) = ( 1st ` F ) ) |
53 |
11
|
ad2antrr |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> D e. Cat ) |
54 |
15
|
ad2antrr |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> E e. Cat ) |
55 |
|
relfunc |
|- Rel ( C Func ( D Xc. E ) ) |
56 |
1 4 2 3
|
prfcl |
|- ( ph -> P e. ( C Func ( D Xc. E ) ) ) |
57 |
|
1st2ndbr |
|- ( ( Rel ( C Func ( D Xc. E ) ) /\ P e. ( C Func ( D Xc. E ) ) ) -> ( 1st ` P ) ( C Func ( D Xc. E ) ) ( 2nd ` P ) ) |
58 |
55 56 57
|
sylancr |
|- ( ph -> ( 1st ` P ) ( C Func ( D Xc. E ) ) ( 2nd ` P ) ) |
59 |
18 7 58
|
funcf1 |
|- ( ph -> ( 1st ` P ) : ( Base ` C ) --> ( ( Base ` D ) X. ( Base ` E ) ) ) |
60 |
59
|
ffvelrnda |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` P ) ` x ) e. ( ( Base ` D ) X. ( Base ` E ) ) ) |
61 |
60
|
adantrr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` P ) ` x ) e. ( ( Base ` D ) X. ( Base ` E ) ) ) |
62 |
61
|
adantr |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( 1st ` P ) ` x ) e. ( ( Base ` D ) X. ( Base ` E ) ) ) |
63 |
59
|
ffvelrnda |
|- ( ( ph /\ y e. ( Base ` C ) ) -> ( ( 1st ` P ) ` y ) e. ( ( Base ` D ) X. ( Base ` E ) ) ) |
64 |
63
|
adantrl |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` P ) ` y ) e. ( ( Base ` D ) X. ( Base ` E ) ) ) |
65 |
64
|
adantr |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( 1st ` P ) ` y ) e. ( ( Base ` D ) X. ( Base ` E ) ) ) |
66 |
4 7 8 53 54 17 62 65
|
1stf2 |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) = ( 1st |` ( ( ( 1st ` P ) ` x ) ( Hom ` ( D Xc. E ) ) ( ( 1st ` P ) ` y ) ) ) ) |
67 |
66
|
fveq1d |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) = ( ( 1st |` ( ( ( 1st ` P ) ` x ) ( Hom ` ( D Xc. E ) ) ( ( 1st ` P ) ` y ) ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) ) |
68 |
58
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` P ) ( C Func ( D Xc. E ) ) ( 2nd ` P ) ) |
69 |
|
simprl |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> x e. ( Base ` C ) ) |
70 |
|
simprr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> y e. ( Base ` C ) ) |
71 |
18 36 8 68 69 70
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` P ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` P ) ` x ) ( Hom ` ( D Xc. E ) ) ( ( 1st ` P ) ` y ) ) ) |
72 |
71
|
ffvelrnda |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` P ) y ) ` f ) e. ( ( ( 1st ` P ) ` x ) ( Hom ` ( D Xc. E ) ) ( ( 1st ` P ) ` y ) ) ) |
73 |
72
|
fvresd |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( 1st |` ( ( ( 1st ` P ) ` x ) ( Hom ` ( D Xc. E ) ) ( ( 1st ` P ) ` y ) ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) = ( 1st ` ( ( x ( 2nd ` P ) y ) ` f ) ) ) |
74 |
2
|
ad2antrr |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> F e. ( C Func D ) ) |
75 |
3
|
ad2antrr |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> G e. ( C Func E ) ) |
76 |
69
|
adantr |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> x e. ( Base ` C ) ) |
77 |
70
|
adantr |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> y e. ( Base ` C ) ) |
78 |
|
simpr |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> f e. ( x ( Hom ` C ) y ) ) |
79 |
1 18 36 74 75 76 77 78
|
prf2 |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` P ) y ) ` f ) = <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. ) |
80 |
79
|
fveq2d |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( 1st ` ( ( x ( 2nd ` P ) y ) ` f ) ) = ( 1st ` <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. ) ) |
81 |
|
fvex |
|- ( ( x ( 2nd ` F ) y ) ` f ) e. _V |
82 |
|
fvex |
|- ( ( x ( 2nd ` G ) y ) ` f ) e. _V |
83 |
81 82
|
op1st |
|- ( 1st ` <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. ) = ( ( x ( 2nd ` F ) y ) ` f ) |
84 |
80 83
|
eqtrdi |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( 1st ` ( ( x ( 2nd ` P ) y ) ` f ) ) = ( ( x ( 2nd ` F ) y ) ` f ) ) |
85 |
67 73 84
|
3eqtrd |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) = ( ( x ( 2nd ` F ) y ) ` f ) ) |
86 |
85
|
mpteq2dva |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( f e. ( x ( Hom ` C ) y ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) ) = ( f e. ( x ( Hom ` C ) y ) |-> ( ( x ( 2nd ` F ) y ) ` f ) ) ) |
87 |
|
eqid |
|- ( Hom ` D ) = ( Hom ` D ) |
88 |
46
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` ( D 1stF E ) ) ( ( D Xc. E ) Func D ) ( 2nd ` ( D 1stF E ) ) ) |
89 |
7 8 87 88 61 64
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) : ( ( ( 1st ` P ) ` x ) ( Hom ` ( D Xc. E ) ) ( ( 1st ` P ) ` y ) ) --> ( ( ( 1st ` ( D 1stF E ) ) ` ( ( 1st ` P ) ` x ) ) ( Hom ` D ) ( ( 1st ` ( D 1stF E ) ) ` ( ( 1st ` P ) ` y ) ) ) ) |
90 |
|
fcompt |
|- ( ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) : ( ( ( 1st ` P ) ` x ) ( Hom ` ( D Xc. E ) ) ( ( 1st ` P ) ` y ) ) --> ( ( ( 1st ` ( D 1stF E ) ) ` ( ( 1st ` P ) ` x ) ) ( Hom ` D ) ( ( 1st ` ( D 1stF E ) ) ` ( ( 1st ` P ) ` y ) ) ) /\ ( x ( 2nd ` P ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` P ) ` x ) ( Hom ` ( D Xc. E ) ) ( ( 1st ` P ) ` y ) ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) = ( f e. ( x ( Hom ` C ) y ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) ) ) |
91 |
89 71 90
|
syl2anc |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) = ( f e. ( x ( Hom ` C ) y ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) ` ( ( x ( 2nd ` P ) y ) ` f ) ) ) ) |
92 |
21
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) ) |
93 |
18 36 87 92 69 70
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) |
94 |
93
|
feqmptd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` F ) y ) = ( f e. ( x ( Hom ` C ) y ) |-> ( ( x ( 2nd ` F ) y ) ` f ) ) ) |
95 |
86 91 94
|
3eqtr4d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) = ( x ( 2nd ` F ) y ) ) |
96 |
95
|
3impb |
|- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) = ( x ( 2nd ` F ) y ) ) |
97 |
96
|
mpoeq3dva |
|- ( ph -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` F ) y ) ) ) |
98 |
18 21
|
funcfn2 |
|- ( ph -> ( 2nd ` F ) Fn ( ( Base ` C ) X. ( Base ` C ) ) ) |
99 |
|
fnov |
|- ( ( 2nd ` F ) Fn ( ( Base ` C ) X. ( Base ` C ) ) <-> ( 2nd ` F ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` F ) y ) ) ) |
100 |
98 99
|
sylib |
|- ( ph -> ( 2nd ` F ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` F ) y ) ) ) |
101 |
97 100
|
eqtr4d |
|- ( ph -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) ) = ( 2nd ` F ) ) |
102 |
52 101
|
opeq12d |
|- ( ph -> <. ( ( 1st ` ( D 1stF E ) ) o. ( 1st ` P ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) ) >. = <. ( 1st ` F ) , ( 2nd ` F ) >. ) |
103 |
18 56 44
|
cofuval |
|- ( ph -> ( ( D 1stF E ) o.func P ) = <. ( ( 1st ` ( D 1stF E ) ) o. ( 1st ` P ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` P ) ` x ) ( 2nd ` ( D 1stF E ) ) ( ( 1st ` P ) ` y ) ) o. ( x ( 2nd ` P ) y ) ) ) >. ) |
104 |
|
1st2nd |
|- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. ) |
105 |
19 2 104
|
sylancr |
|- ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. ) |
106 |
102 103 105
|
3eqtr4d |
|- ( ph -> ( ( D 1stF E ) o.func P ) = F ) |