Step |
Hyp |
Ref |
Expression |
1 |
|
natfval.1 |
|- N = ( C Nat D ) |
2 |
|
natfval.b |
|- B = ( Base ` C ) |
3 |
|
natfval.h |
|- H = ( Hom ` C ) |
4 |
|
natfval.j |
|- J = ( Hom ` D ) |
5 |
|
natfval.o |
|- .x. = ( comp ` D ) |
6 |
|
isnat.f |
|- ( ph -> F ( C Func D ) G ) |
7 |
|
isnat.g |
|- ( ph -> K ( C Func D ) L ) |
8 |
1 2 3 4 5
|
natfval |
|- N = ( f e. ( C Func D ) , g e. ( C Func D ) |-> [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. B ( ( r ` x ) J ( s ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. .x. ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. .x. ( s ` y ) ) ( a ` x ) ) } ) |
9 |
8
|
a1i |
|- ( ph -> N = ( f e. ( C Func D ) , g e. ( C Func D ) |-> [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. B ( ( r ` x ) J ( s ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. .x. ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. .x. ( s ` y ) ) ( a ` x ) ) } ) ) |
10 |
|
fvexd |
|- ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) -> ( 1st ` f ) e. _V ) |
11 |
|
simprl |
|- ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) -> f = <. F , G >. ) |
12 |
11
|
fveq2d |
|- ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) -> ( 1st ` f ) = ( 1st ` <. F , G >. ) ) |
13 |
|
relfunc |
|- Rel ( C Func D ) |
14 |
|
brrelex12 |
|- ( ( Rel ( C Func D ) /\ F ( C Func D ) G ) -> ( F e. _V /\ G e. _V ) ) |
15 |
13 6 14
|
sylancr |
|- ( ph -> ( F e. _V /\ G e. _V ) ) |
16 |
|
op1stg |
|- ( ( F e. _V /\ G e. _V ) -> ( 1st ` <. F , G >. ) = F ) |
17 |
15 16
|
syl |
|- ( ph -> ( 1st ` <. F , G >. ) = F ) |
18 |
17
|
adantr |
|- ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) -> ( 1st ` <. F , G >. ) = F ) |
19 |
12 18
|
eqtrd |
|- ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) -> ( 1st ` f ) = F ) |
20 |
|
fvexd |
|- ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) -> ( 1st ` g ) e. _V ) |
21 |
|
simplrr |
|- ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) -> g = <. K , L >. ) |
22 |
21
|
fveq2d |
|- ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) -> ( 1st ` g ) = ( 1st ` <. K , L >. ) ) |
23 |
|
brrelex12 |
|- ( ( Rel ( C Func D ) /\ K ( C Func D ) L ) -> ( K e. _V /\ L e. _V ) ) |
24 |
13 7 23
|
sylancr |
|- ( ph -> ( K e. _V /\ L e. _V ) ) |
25 |
|
op1stg |
|- ( ( K e. _V /\ L e. _V ) -> ( 1st ` <. K , L >. ) = K ) |
26 |
24 25
|
syl |
|- ( ph -> ( 1st ` <. K , L >. ) = K ) |
27 |
26
|
ad2antrr |
|- ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) -> ( 1st ` <. K , L >. ) = K ) |
28 |
22 27
|
eqtrd |
|- ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) -> ( 1st ` g ) = K ) |
29 |
|
simplr |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> r = F ) |
30 |
29
|
fveq1d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( r ` x ) = ( F ` x ) ) |
31 |
|
simpr |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> s = K ) |
32 |
31
|
fveq1d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( s ` x ) = ( K ` x ) ) |
33 |
30 32
|
oveq12d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( ( r ` x ) J ( s ` x ) ) = ( ( F ` x ) J ( K ` x ) ) ) |
34 |
33
|
ixpeq2dv |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> X_ x e. B ( ( r ` x ) J ( s ` x ) ) = X_ x e. B ( ( F ` x ) J ( K ` x ) ) ) |
35 |
29
|
fveq1d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( r ` y ) = ( F ` y ) ) |
36 |
30 35
|
opeq12d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> <. ( r ` x ) , ( r ` y ) >. = <. ( F ` x ) , ( F ` y ) >. ) |
37 |
31
|
fveq1d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( s ` y ) = ( K ` y ) ) |
38 |
36 37
|
oveq12d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( <. ( r ` x ) , ( r ` y ) >. .x. ( s ` y ) ) = ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ) |
39 |
|
eqidd |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( a ` y ) = ( a ` y ) ) |
40 |
11
|
ad2antrr |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> f = <. F , G >. ) |
41 |
40
|
fveq2d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( 2nd ` f ) = ( 2nd ` <. F , G >. ) ) |
42 |
|
op2ndg |
|- ( ( F e. _V /\ G e. _V ) -> ( 2nd ` <. F , G >. ) = G ) |
43 |
15 42
|
syl |
|- ( ph -> ( 2nd ` <. F , G >. ) = G ) |
44 |
43
|
ad3antrrr |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( 2nd ` <. F , G >. ) = G ) |
45 |
41 44
|
eqtrd |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( 2nd ` f ) = G ) |
46 |
45
|
oveqd |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( x ( 2nd ` f ) y ) = ( x G y ) ) |
47 |
46
|
fveq1d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( ( x ( 2nd ` f ) y ) ` h ) = ( ( x G y ) ` h ) ) |
48 |
38 39 47
|
oveq123d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. .x. ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) ) |
49 |
30 32
|
opeq12d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> <. ( r ` x ) , ( s ` x ) >. = <. ( F ` x ) , ( K ` x ) >. ) |
50 |
49 37
|
oveq12d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( <. ( r ` x ) , ( s ` x ) >. .x. ( s ` y ) ) = ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ) |
51 |
21
|
adantr |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> g = <. K , L >. ) |
52 |
51
|
fveq2d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( 2nd ` g ) = ( 2nd ` <. K , L >. ) ) |
53 |
|
op2ndg |
|- ( ( K e. _V /\ L e. _V ) -> ( 2nd ` <. K , L >. ) = L ) |
54 |
24 53
|
syl |
|- ( ph -> ( 2nd ` <. K , L >. ) = L ) |
55 |
54
|
ad3antrrr |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( 2nd ` <. K , L >. ) = L ) |
56 |
52 55
|
eqtrd |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( 2nd ` g ) = L ) |
57 |
56
|
oveqd |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( x ( 2nd ` g ) y ) = ( x L y ) ) |
58 |
57
|
fveq1d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( ( x ( 2nd ` g ) y ) ` h ) = ( ( x L y ) ` h ) ) |
59 |
|
eqidd |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( a ` x ) = ( a ` x ) ) |
60 |
50 58 59
|
oveq123d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. .x. ( s ` y ) ) ( a ` x ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) ) |
61 |
48 60
|
eqeq12d |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. .x. ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. .x. ( s ` y ) ) ( a ` x ) ) <-> ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) ) ) |
62 |
61
|
ralbidv |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( A. h e. ( x H y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. .x. ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. .x. ( s ` y ) ) ( a ` x ) ) <-> A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) ) ) |
63 |
62
|
2ralbidv |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> ( A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. .x. ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. .x. ( s ` y ) ) ( a ` x ) ) <-> A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) ) ) |
64 |
34 63
|
rabeqbidv |
|- ( ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) /\ s = K ) -> { a e. X_ x e. B ( ( r ` x ) J ( s ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. .x. ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. .x. ( s ` y ) ) ( a ` x ) ) } = { a e. X_ x e. B ( ( F ` x ) J ( K ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) } ) |
65 |
20 28 64
|
csbied2 |
|- ( ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) /\ r = F ) -> [_ ( 1st ` g ) / s ]_ { a e. X_ x e. B ( ( r ` x ) J ( s ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. .x. ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. .x. ( s ` y ) ) ( a ` x ) ) } = { a e. X_ x e. B ( ( F ` x ) J ( K ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) } ) |
66 |
10 19 65
|
csbied2 |
|- ( ( ph /\ ( f = <. F , G >. /\ g = <. K , L >. ) ) -> [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. B ( ( r ` x ) J ( s ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. .x. ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. .x. ( s ` y ) ) ( a ` x ) ) } = { a e. X_ x e. B ( ( F ` x ) J ( K ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) } ) |
67 |
|
df-br |
|- ( F ( C Func D ) G <-> <. F , G >. e. ( C Func D ) ) |
68 |
6 67
|
sylib |
|- ( ph -> <. F , G >. e. ( C Func D ) ) |
69 |
|
df-br |
|- ( K ( C Func D ) L <-> <. K , L >. e. ( C Func D ) ) |
70 |
7 69
|
sylib |
|- ( ph -> <. K , L >. e. ( C Func D ) ) |
71 |
|
ovex |
|- ( ( F ` x ) J ( K ` x ) ) e. _V |
72 |
71
|
rgenw |
|- A. x e. B ( ( F ` x ) J ( K ` x ) ) e. _V |
73 |
|
ixpexg |
|- ( A. x e. B ( ( F ` x ) J ( K ` x ) ) e. _V -> X_ x e. B ( ( F ` x ) J ( K ` x ) ) e. _V ) |
74 |
72 73
|
ax-mp |
|- X_ x e. B ( ( F ` x ) J ( K ` x ) ) e. _V |
75 |
74
|
rabex |
|- { a e. X_ x e. B ( ( F ` x ) J ( K ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) } e. _V |
76 |
75
|
a1i |
|- ( ph -> { a e. X_ x e. B ( ( F ` x ) J ( K ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) } e. _V ) |
77 |
9 66 68 70 76
|
ovmpod |
|- ( ph -> ( <. F , G >. N <. K , L >. ) = { a e. X_ x e. B ( ( F ` x ) J ( K ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) } ) |
78 |
77
|
eleq2d |
|- ( ph -> ( A e. ( <. F , G >. N <. K , L >. ) <-> A e. { a e. X_ x e. B ( ( F ` x ) J ( K ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) } ) ) |
79 |
|
fveq1 |
|- ( a = A -> ( a ` y ) = ( A ` y ) ) |
80 |
79
|
oveq1d |
|- ( a = A -> ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( A ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) ) |
81 |
|
fveq1 |
|- ( a = A -> ( a ` x ) = ( A ` x ) ) |
82 |
81
|
oveq2d |
|- ( a = A -> ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( A ` x ) ) ) |
83 |
80 82
|
eqeq12d |
|- ( a = A -> ( ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) <-> ( ( A ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( A ` x ) ) ) ) |
84 |
83
|
ralbidv |
|- ( a = A -> ( A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) <-> A. h e. ( x H y ) ( ( A ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( A ` x ) ) ) ) |
85 |
84
|
2ralbidv |
|- ( a = A -> ( A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) <-> A. x e. B A. y e. B A. h e. ( x H y ) ( ( A ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( A ` x ) ) ) ) |
86 |
85
|
elrab |
|- ( A e. { a e. X_ x e. B ( ( F ` x ) J ( K ` x ) ) | A. x e. B A. y e. B A. h e. ( x H y ) ( ( a ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( a ` x ) ) } <-> ( A e. X_ x e. B ( ( F ` x ) J ( K ` x ) ) /\ A. x e. B A. y e. B A. h e. ( x H y ) ( ( A ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( A ` x ) ) ) ) |
87 |
78 86
|
bitrdi |
|- ( ph -> ( A e. ( <. F , G >. N <. K , L >. ) <-> ( A e. X_ x e. B ( ( F ` x ) J ( K ` x ) ) /\ A. x e. B A. y e. B A. h e. ( x H y ) ( ( A ` y ) ( <. ( F ` x ) , ( F ` y ) >. .x. ( K ` y ) ) ( ( x G y ) ` h ) ) = ( ( ( x L y ) ` h ) ( <. ( F ` x ) , ( K ` x ) >. .x. ( K ` y ) ) ( A ` x ) ) ) ) ) |