Step |
Hyp |
Ref |
Expression |
1 |
|
fprodcnv.1 |
|- ( x = <. j , k >. -> B = D ) |
2 |
|
fprodcnv.2 |
|- ( y = <. k , j >. -> C = D ) |
3 |
|
fprodcnv.3 |
|- ( ph -> A e. Fin ) |
4 |
|
fprodcnv.4 |
|- ( ph -> Rel A ) |
5 |
|
fprodcnv.5 |
|- ( ( ph /\ x e. A ) -> B e. CC ) |
6 |
|
csbeq1a |
|- ( x = <. ( 2nd ` y ) , ( 1st ` y ) >. -> B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B ) |
7 |
|
fvex |
|- ( 2nd ` y ) e. _V |
8 |
|
fvex |
|- ( 1st ` y ) e. _V |
9 |
|
opex |
|- <. j , k >. e. _V |
10 |
9 1
|
csbie |
|- [_ <. j , k >. / x ]_ B = D |
11 |
|
opeq12 |
|- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> <. j , k >. = <. ( 2nd ` y ) , ( 1st ` y ) >. ) |
12 |
11
|
csbeq1d |
|- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. j , k >. / x ]_ B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B ) |
13 |
10 12
|
eqtr3id |
|- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B ) |
14 |
7 8 13
|
csbie2 |
|- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B |
15 |
6 14
|
eqtr4di |
|- ( x = <. ( 2nd ` y ) , ( 1st ` y ) >. -> B = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D ) |
16 |
|
cnvfi |
|- ( A e. Fin -> `' A e. Fin ) |
17 |
3 16
|
syl |
|- ( ph -> `' A e. Fin ) |
18 |
|
relcnv |
|- Rel `' A |
19 |
|
cnvf1o |
|- ( Rel `' A -> ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> `' `' A ) |
20 |
18 19
|
ax-mp |
|- ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> `' `' A |
21 |
|
dfrel2 |
|- ( Rel A <-> `' `' A = A ) |
22 |
4 21
|
sylib |
|- ( ph -> `' `' A = A ) |
23 |
22
|
f1oeq3d |
|- ( ph -> ( ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> `' `' A <-> ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> A ) ) |
24 |
20 23
|
mpbii |
|- ( ph -> ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> A ) |
25 |
|
1st2nd |
|- ( ( Rel `' A /\ y e. `' A ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. ) |
26 |
18 25
|
mpan |
|- ( y e. `' A -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. ) |
27 |
26
|
fveq2d |
|- ( y e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` y ) = ( ( z e. `' A |-> U. `' { z } ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) ) |
28 |
26
|
eleq1d |
|- ( y e. `' A -> ( y e. `' A <-> <. ( 1st ` y ) , ( 2nd ` y ) >. e. `' A ) ) |
29 |
28
|
ibi |
|- ( y e. `' A -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. `' A ) |
30 |
|
sneq |
|- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> { z } = { <. ( 1st ` y ) , ( 2nd ` y ) >. } ) |
31 |
30
|
cnveqd |
|- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> `' { z } = `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } ) |
32 |
31
|
unieqd |
|- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } ) |
33 |
|
opswap |
|- U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } = <. ( 2nd ` y ) , ( 1st ` y ) >. |
34 |
32 33
|
eqtrdi |
|- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = <. ( 2nd ` y ) , ( 1st ` y ) >. ) |
35 |
|
eqid |
|- ( z e. `' A |-> U. `' { z } ) = ( z e. `' A |-> U. `' { z } ) |
36 |
|
opex |
|- <. ( 2nd ` y ) , ( 1st ` y ) >. e. _V |
37 |
34 35 36
|
fvmpt |
|- ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) = <. ( 2nd ` y ) , ( 1st ` y ) >. ) |
38 |
29 37
|
syl |
|- ( y e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) = <. ( 2nd ` y ) , ( 1st ` y ) >. ) |
39 |
27 38
|
eqtrd |
|- ( y e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` y ) = <. ( 2nd ` y ) , ( 1st ` y ) >. ) |
40 |
39
|
adantl |
|- ( ( ph /\ y e. `' A ) -> ( ( z e. `' A |-> U. `' { z } ) ` y ) = <. ( 2nd ` y ) , ( 1st ` y ) >. ) |
41 |
15 17 24 40 5
|
fprodf1o |
|- ( ph -> prod_ x e. A B = prod_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D ) |
42 |
|
csbeq1a |
|- ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. -> C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C ) |
43 |
26 42
|
syl |
|- ( y e. `' A -> C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C ) |
44 |
|
opex |
|- <. k , j >. e. _V |
45 |
44 2
|
csbie |
|- [_ <. k , j >. / y ]_ C = D |
46 |
|
opeq12 |
|- ( ( k = ( 1st ` y ) /\ j = ( 2nd ` y ) ) -> <. k , j >. = <. ( 1st ` y ) , ( 2nd ` y ) >. ) |
47 |
46
|
ancoms |
|- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> <. k , j >. = <. ( 1st ` y ) , ( 2nd ` y ) >. ) |
48 |
47
|
csbeq1d |
|- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. k , j >. / y ]_ C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C ) |
49 |
45 48
|
eqtr3id |
|- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C ) |
50 |
7 8 49
|
csbie2 |
|- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C |
51 |
43 50
|
eqtr4di |
|- ( y e. `' A -> C = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D ) |
52 |
51
|
prodeq2i |
|- prod_ y e. `' A C = prod_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D |
53 |
41 52
|
eqtr4di |
|- ( ph -> prod_ x e. A B = prod_ y e. `' A C ) |