ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  genipv Unicode version

Theorem genipv 7499
Description: Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)
Hypotheses
Ref Expression
genp.1  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
genp.2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
Assertion
Ref Expression
genipv  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  =  <. { q  e.  Q.  |  E. r  e.  ( 1st `  A ) E. s  e.  ( 1st `  B
) q  =  ( r G s ) } ,  { q  e.  Q.  |  E. r  e.  ( 2nd `  A ) E. s  e.  ( 2nd `  B
) q  =  ( r G s ) } >. )
Distinct variable groups:    x, y, z, q, r, s, A   
x, B, y, z, q, r, s    x, w, v, G, y, z, q, r, s
Allowed substitution hints:    A( w, v)    B( w, v)    F( x, y, z, w, v, s, r, q)

Proof of Theorem genipv
Dummy variables  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5876 . . . 4  |-  ( f  =  A  ->  (
f F g )  =  ( A F g ) )
2 fveq2 5511 . . . . . . 7  |-  ( f  =  A  ->  ( 1st `  f )  =  ( 1st `  A
) )
32rexeqdv 2679 . . . . . 6  |-  ( f  =  A  ->  ( E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g ) x  =  ( y G z )  <->  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) ) )
43rabbidv 2726 . . . . 5  |-  ( f  =  A  ->  { x  e.  Q.  |  E. y  e.  ( 1st `  f
) E. z  e.  ( 1st `  g
) x  =  ( y G z ) }  =  { x  e.  Q.  |  E. y  e.  ( 1st `  A
) E. z  e.  ( 1st `  g
) x  =  ( y G z ) } )
5 fveq2 5511 . . . . . . 7  |-  ( f  =  A  ->  ( 2nd `  f )  =  ( 2nd `  A
) )
65rexeqdv 2679 . . . . . 6  |-  ( f  =  A  ->  ( E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  g ) x  =  ( y G z )  <->  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  g ) x  =  ( y G z ) ) )
76rabbidv 2726 . . . . 5  |-  ( f  =  A  ->  { x  e.  Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) }  =  { x  e.  Q.  |  E. y  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) } )
84, 7opeq12d 3784 . . . 4  |-  ( f  =  A  ->  <. { x  e.  Q.  |  E. y  e.  ( 1st `  f
) E. z  e.  ( 1st `  g
) x  =  ( y G z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) } >.  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) } ,  {
x  e.  Q.  |  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  g ) x  =  ( y G z ) } >. )
91, 8eqeq12d 2192 . . 3  |-  ( f  =  A  ->  (
( f F g )  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  f
) E. z  e.  ( 1st `  g
) x  =  ( y G z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) } >.  <->  ( A F g )  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) } ,  {
x  e.  Q.  |  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  g ) x  =  ( y G z ) } >. )
)
10 oveq2 5877 . . . 4  |-  ( g  =  B  ->  ( A F g )  =  ( A F B ) )
11 fveq2 5511 . . . . . . . 8  |-  ( g  =  B  ->  ( 1st `  g )  =  ( 1st `  B
) )
1211rexeqdv 2679 . . . . . . 7  |-  ( g  =  B  ->  ( E. z  e.  ( 1st `  g ) x  =  ( y G z )  <->  E. z  e.  ( 1st `  B
) x  =  ( y G z ) ) )
1312rexbidv 2478 . . . . . 6  |-  ( g  =  B  ->  ( E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  g ) x  =  ( y G z )  <->  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B ) x  =  ( y G z ) ) )
1413rabbidv 2726 . . . . 5  |-  ( g  =  B  ->  { x  e.  Q.  |  E. y  e.  ( 1st `  A
) E. z  e.  ( 1st `  g
) x  =  ( y G z ) }  =  { x  e.  Q.  |  E. y  e.  ( 1st `  A
) E. z  e.  ( 1st `  B
) x  =  ( y G z ) } )
15 fveq2 5511 . . . . . . . 8  |-  ( g  =  B  ->  ( 2nd `  g )  =  ( 2nd `  B
) )
1615rexeqdv 2679 . . . . . . 7  |-  ( g  =  B  ->  ( E. z  e.  ( 2nd `  g ) x  =  ( y G z )  <->  E. z  e.  ( 2nd `  B
) x  =  ( y G z ) ) )
1716rexbidv 2478 . . . . . 6  |-  ( g  =  B  ->  ( E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  g ) x  =  ( y G z )  <->  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B ) x  =  ( y G z ) ) )
1817rabbidv 2726 . . . . 5  |-  ( g  =  B  ->  { x  e.  Q.  |  E. y  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) }  =  { x  e.  Q.  |  E. y  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  B
) x  =  ( y G z ) } )
1914, 18opeq12d 3784 . . . 4  |-  ( g  =  B  ->  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A
) E. z  e.  ( 1st `  g
) x  =  ( y G z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) } >.  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B ) x  =  ( y G z ) } ,  {
x  e.  Q.  |  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B ) x  =  ( y G z ) } >. )
2010, 19eqeq12d 2192 . . 3  |-  ( g  =  B  ->  (
( A F g )  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A
) E. z  e.  ( 1st `  g
) x  =  ( y G z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) } >.  <->  ( A F B )  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B ) x  =  ( y G z ) } ,  {
x  e.  Q.  |  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B ) x  =  ( y G z ) } >. )
)
21 nqex 7353 . . . . . . 7  |-  Q.  e.  _V
2221a1i 9 . . . . . 6  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  Q.  e.  _V )
23 rabssab 3243 . . . . . . 7  |-  { x  e.  Q.  |  E. y  e.  ( 1st `  f
) E. z  e.  ( 1st `  g
) x  =  ( y G z ) }  C_  { x  |  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) }
24 prop 7465 . . . . . . . . . . . 12  |-  ( f  e.  P.  ->  <. ( 1st `  f ) ,  ( 2nd `  f
) >.  e.  P. )
25 elprnql 7471 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  f
) ,  ( 2nd `  f ) >.  e.  P.  /\  y  e.  ( 1st `  f ) )  -> 
y  e.  Q. )
2624, 25sylan 283 . . . . . . . . . . 11  |-  ( ( f  e.  P.  /\  y  e.  ( 1st `  f ) )  -> 
y  e.  Q. )
27 prop 7465 . . . . . . . . . . . 12  |-  ( g  e.  P.  ->  <. ( 1st `  g ) ,  ( 2nd `  g
) >.  e.  P. )
28 elprnql 7471 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  g
) ,  ( 2nd `  g ) >.  e.  P.  /\  z  e.  ( 1st `  g ) )  -> 
z  e.  Q. )
2927, 28sylan 283 . . . . . . . . . . 11  |-  ( ( g  e.  P.  /\  z  e.  ( 1st `  g ) )  -> 
z  e.  Q. )
30 genp.2 . . . . . . . . . . . 12  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
31 eleq1 2240 . . . . . . . . . . . 12  |-  ( x  =  ( y G z )  ->  (
x  e.  Q.  <->  ( y G z )  e. 
Q. ) )
3230, 31syl5ibrcom 157 . . . . . . . . . . 11  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( x  =  ( y G z )  ->  x  e.  Q. ) )
3326, 29, 32syl2an 289 . . . . . . . . . 10  |-  ( ( ( f  e.  P.  /\  y  e.  ( 1st `  f ) )  /\  ( g  e.  P.  /\  z  e.  ( 1st `  g ) ) )  ->  ( x  =  ( y G z )  ->  x  e.  Q. ) )
3433an4s 588 . . . . . . . . 9  |-  ( ( ( f  e.  P.  /\  g  e.  P. )  /\  ( y  e.  ( 1st `  f )  /\  z  e.  ( 1st `  g ) ) )  ->  (
x  =  ( y G z )  ->  x  e.  Q. )
)
3534rexlimdvva 2602 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( E. y  e.  ( 1st `  f
) E. z  e.  ( 1st `  g
) x  =  ( y G z )  ->  x  e.  Q. ) )
3635abssdv 3229 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  { x  |  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g
) x  =  ( y G z ) }  C_  Q. )
3723, 36sstrid 3166 . . . . . 6  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  { x  e.  Q.  |  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) } 
C_  Q. )
3822, 37ssexd 4140 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  { x  e.  Q.  |  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) }  e.  _V )
39 rabssab 3243 . . . . . . 7  |-  { x  e.  Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) }  C_  { x  |  E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  g ) x  =  ( y G z ) }
40 elprnqu 7472 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  f
) ,  ( 2nd `  f ) >.  e.  P.  /\  y  e.  ( 2nd `  f ) )  -> 
y  e.  Q. )
4124, 40sylan 283 . . . . . . . . . . 11  |-  ( ( f  e.  P.  /\  y  e.  ( 2nd `  f ) )  -> 
y  e.  Q. )
42 elprnqu 7472 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  g
) ,  ( 2nd `  g ) >.  e.  P.  /\  z  e.  ( 2nd `  g ) )  -> 
z  e.  Q. )
4327, 42sylan 283 . . . . . . . . . . 11  |-  ( ( g  e.  P.  /\  z  e.  ( 2nd `  g ) )  -> 
z  e.  Q. )
4441, 43, 32syl2an 289 . . . . . . . . . 10  |-  ( ( ( f  e.  P.  /\  y  e.  ( 2nd `  f ) )  /\  ( g  e.  P.  /\  z  e.  ( 2nd `  g ) ) )  ->  ( x  =  ( y G z )  ->  x  e.  Q. ) )
4544an4s 588 . . . . . . . . 9  |-  ( ( ( f  e.  P.  /\  g  e.  P. )  /\  ( y  e.  ( 2nd `  f )  /\  z  e.  ( 2nd `  g ) ) )  ->  (
x  =  ( y G z )  ->  x  e.  Q. )
)
4645rexlimdvva 2602 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  g
) x  =  ( y G z )  ->  x  e.  Q. ) )
4746abssdv 3229 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  { x  |  E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) }  C_  Q. )
4839, 47sstrid 3166 . . . . . 6  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  { x  e.  Q.  |  E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  g ) x  =  ( y G z ) } 
C_  Q. )
4922, 48ssexd 4140 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  { x  e.  Q.  |  E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  g ) x  =  ( y G z ) }  e.  _V )
50 opelxp 4653 . . . . 5  |-  ( <. { x  e.  Q.  |  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) } ,  { x  e. 
Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) } >.  e.  ( _V  X.  _V )  <->  ( {
x  e.  Q.  |  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) }  e.  _V  /\ 
{ x  e.  Q.  |  E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  g ) x  =  ( y G z ) }  e.  _V ) )
5138, 49, 50sylanbrc 417 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  -> 
<. { x  e.  Q.  |  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) } ,  { x  e. 
Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) } >.  e.  ( _V  X.  _V ) )
52 fveq2 5511 . . . . . . . 8  |-  ( w  =  f  ->  ( 1st `  w )  =  ( 1st `  f
) )
5352rexeqdv 2679 . . . . . . 7  |-  ( w  =  f  ->  ( E. y  e.  ( 1st `  w ) E. z  e.  ( 1st `  v ) x  =  ( y G z )  <->  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  v ) x  =  ( y G z ) ) )
5453rabbidv 2726 . . . . . 6  |-  ( w  =  f  ->  { x  e.  Q.  |  E. y  e.  ( 1st `  w
) E. z  e.  ( 1st `  v
) x  =  ( y G z ) }  =  { x  e.  Q.  |  E. y  e.  ( 1st `  f
) E. z  e.  ( 1st `  v
) x  =  ( y G z ) } )
55 fveq2 5511 . . . . . . . 8  |-  ( w  =  f  ->  ( 2nd `  w )  =  ( 2nd `  f
) )
5655rexeqdv 2679 . . . . . . 7  |-  ( w  =  f  ->  ( E. y  e.  ( 2nd `  w ) E. z  e.  ( 2nd `  v ) x  =  ( y G z )  <->  E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  v ) x  =  ( y G z ) ) )
5756rabbidv 2726 . . . . . 6  |-  ( w  =  f  ->  { x  e.  Q.  |  E. y  e.  ( 2nd `  w
) E. z  e.  ( 2nd `  v
) x  =  ( y G z ) }  =  { x  e.  Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  v
) x  =  ( y G z ) } )
5854, 57opeq12d 3784 . . . . 5  |-  ( w  =  f  ->  <. { x  e.  Q.  |  E. y  e.  ( 1st `  w
) E. z  e.  ( 1st `  v
) x  =  ( y G z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  w
) E. z  e.  ( 2nd `  v
) x  =  ( y G z ) } >.  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  v ) x  =  ( y G z ) } ,  {
x  e.  Q.  |  E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  v ) x  =  ( y G z ) } >. )
59 fveq2 5511 . . . . . . . . 9  |-  ( v  =  g  ->  ( 1st `  v )  =  ( 1st `  g
) )
6059rexeqdv 2679 . . . . . . . 8  |-  ( v  =  g  ->  ( E. z  e.  ( 1st `  v ) x  =  ( y G z )  <->  E. z  e.  ( 1st `  g
) x  =  ( y G z ) ) )
6160rexbidv 2478 . . . . . . 7  |-  ( v  =  g  ->  ( E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  v ) x  =  ( y G z )  <->  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) ) )
6261rabbidv 2726 . . . . . 6  |-  ( v  =  g  ->  { x  e.  Q.  |  E. y  e.  ( 1st `  f
) E. z  e.  ( 1st `  v
) x  =  ( y G z ) }  =  { x  e.  Q.  |  E. y  e.  ( 1st `  f
) E. z  e.  ( 1st `  g
) x  =  ( y G z ) } )
63 fveq2 5511 . . . . . . . . 9  |-  ( v  =  g  ->  ( 2nd `  v )  =  ( 2nd `  g
) )
6463rexeqdv 2679 . . . . . . . 8  |-  ( v  =  g  ->  ( E. z  e.  ( 2nd `  v ) x  =  ( y G z )  <->  E. z  e.  ( 2nd `  g
) x  =  ( y G z ) ) )
6564rexbidv 2478 . . . . . . 7  |-  ( v  =  g  ->  ( E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  v ) x  =  ( y G z )  <->  E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  g ) x  =  ( y G z ) ) )
6665rabbidv 2726 . . . . . 6  |-  ( v  =  g  ->  { x  e.  Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  v
) x  =  ( y G z ) }  =  { x  e.  Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) } )
6762, 66opeq12d 3784 . . . . 5  |-  ( v  =  g  ->  <. { x  e.  Q.  |  E. y  e.  ( 1st `  f
) E. z  e.  ( 1st `  v
) x  =  ( y G z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  v
) x  =  ( y G z ) } >.  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) } ,  {
x  e.  Q.  |  E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  g ) x  =  ( y G z ) } >. )
68 genp.1 . . . . . 6  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
6968genpdf 7498 . . . . 5  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  ( 1st `  w ) E. z  e.  ( 1st `  v ) x  =  ( y G z ) } ,  { x  e. 
Q.  |  E. y  e.  ( 2nd `  w
) E. z  e.  ( 2nd `  v
) x  =  ( y G z ) } >. )
7058, 67, 69ovmpog 6003 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  <. { x  e.  Q.  |  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) } ,  {
x  e.  Q.  |  E. y  e.  ( 2nd `  f ) E. z  e.  ( 2nd `  g ) x  =  ( y G z ) } >.  e.  ( _V  X.  _V )
)  ->  ( f F g )  = 
<. { x  e.  Q.  |  E. y  e.  ( 1st `  f ) E. z  e.  ( 1st `  g ) x  =  ( y G z ) } ,  { x  e. 
Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) } >. )
7151, 70mpd3an3 1338 . . 3  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f F g )  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  f
) E. z  e.  ( 1st `  g
) x  =  ( y G z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  f
) E. z  e.  ( 2nd `  g
) x  =  ( y G z ) } >. )
729, 20, 71vtocl2ga 2805 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A
) E. z  e.  ( 1st `  B
) x  =  ( y G z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  B
) x  =  ( y G z ) } >. )
73 eqeq1 2184 . . . . . 6  |-  ( x  =  q  ->  (
x  =  ( y G z )  <->  q  =  ( y G z ) ) )
74732rexbidv 2502 . . . . 5  |-  ( x  =  q  ->  ( E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B ) x  =  ( y G z )  <->  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B ) q  =  ( y G z ) ) )
75 oveq1 5876 . . . . . . 7  |-  ( y  =  r  ->  (
y G z )  =  ( r G z ) )
7675eqeq2d 2189 . . . . . 6  |-  ( y  =  r  ->  (
q  =  ( y G z )  <->  q  =  ( r G z ) ) )
77 oveq2 5877 . . . . . . 7  |-  ( z  =  s  ->  (
r G z )  =  ( r G s ) )
7877eqeq2d 2189 . . . . . 6  |-  ( z  =  s  ->  (
q  =  ( r G z )  <->  q  =  ( r G s ) ) )
7976, 78cbvrex2v 2717 . . . . 5  |-  ( E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B
) q  =  ( y G z )  <->  E. r  e.  ( 1st `  A ) E. s  e.  ( 1st `  B ) q  =  ( r G s ) )
8074, 79bitrdi 196 . . . 4  |-  ( x  =  q  ->  ( E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B ) x  =  ( y G z )  <->  E. r  e.  ( 1st `  A ) E. s  e.  ( 1st `  B ) q  =  ( r G s ) ) )
8180cbvrabv 2736 . . 3  |-  { x  e.  Q.  |  E. y  e.  ( 1st `  A
) E. z  e.  ( 1st `  B
) x  =  ( y G z ) }  =  { q  e.  Q.  |  E. r  e.  ( 1st `  A ) E. s  e.  ( 1st `  B
) q  =  ( r G s ) }
82732rexbidv 2502 . . . . 5  |-  ( x  =  q  ->  ( E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B ) x  =  ( y G z )  <->  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B ) q  =  ( y G z ) ) )
8376, 78cbvrex2v 2717 . . . . 5  |-  ( E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B
) q  =  ( y G z )  <->  E. r  e.  ( 2nd `  A ) E. s  e.  ( 2nd `  B ) q  =  ( r G s ) )
8482, 83bitrdi 196 . . . 4  |-  ( x  =  q  ->  ( E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B ) x  =  ( y G z )  <->  E. r  e.  ( 2nd `  A ) E. s  e.  ( 2nd `  B ) q  =  ( r G s ) ) )
8584cbvrabv 2736 . . 3  |-  { x  e.  Q.  |  E. y  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  B
) x  =  ( y G z ) }  =  { q  e.  Q.  |  E. r  e.  ( 2nd `  A ) E. s  e.  ( 2nd `  B
) q  =  ( r G s ) }
8681, 85opeq12i 3781 . 2  |-  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A
) E. z  e.  ( 1st `  B
) x  =  ( y G z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  B
) x  =  ( y G z ) } >.  =  <. { q  e.  Q.  |  E. r  e.  ( 1st `  A ) E. s  e.  ( 1st `  B ) q  =  ( r G s ) } ,  {
q  e.  Q.  |  E. r  e.  ( 2nd `  A ) E. s  e.  ( 2nd `  B ) q  =  ( r G s ) } >.
8772, 86eqtrdi 2226 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  =  <. { q  e.  Q.  |  E. r  e.  ( 1st `  A ) E. s  e.  ( 1st `  B
) q  =  ( r G s ) } ,  { q  e.  Q.  |  E. r  e.  ( 2nd `  A ) E. s  e.  ( 2nd `  B
) q  =  ( r G s ) } >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   {cab 2163   E.wrex 2456   {crab 2459   _Vcvv 2737   <.cop 3594    X. cxp 4621   ` cfv 5212  (class class class)co 5869    e. cmpo 5871   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270   P.cnp 7281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-qs 6535  df-ni 7294  df-nqqs 7338  df-inp 7456
This theorem is referenced by:  genpelvl  7502  genpelvu  7503  plpvlu  7528  mpvlu  7529
  Copyright terms: Public domain W3C validator