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Theorem genipv 7285
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 5749 . . . 4  |-  ( f  =  A  ->  (
f F g )  =  ( A F g ) )
2 fveq2 5389 . . . . . . 7  |-  ( f  =  A  ->  ( 1st `  f )  =  ( 1st `  A
) )
32rexeqdv 2610 . . . . . 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 2649 . . . . 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 5389 . . . . . . 7  |-  ( f  =  A  ->  ( 2nd `  f )  =  ( 2nd `  A
) )
65rexeqdv 2610 . . . . . 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 2649 . . . . 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 3683 . . . 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 2132 . . 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 5750 . . . 4  |-  ( g  =  B  ->  ( A F g )  =  ( A F B ) )
11 fveq2 5389 . . . . . . . 8  |-  ( g  =  B  ->  ( 1st `  g )  =  ( 1st `  B
) )
1211rexeqdv 2610 . . . . . . 7  |-  ( g  =  B  ->  ( E. z  e.  ( 1st `  g ) x  =  ( y G z )  <->  E. z  e.  ( 1st `  B
) x  =  ( y G z ) ) )
1312rexbidv 2415 . . . . . 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 2649 . . . . 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 5389 . . . . . . . 8  |-  ( g  =  B  ->  ( 2nd `  g )  =  ( 2nd `  B
) )
1615rexeqdv 2610 . . . . . . 7  |-  ( g  =  B  ->  ( E. z  e.  ( 2nd `  g ) x  =  ( y G z )  <->  E. z  e.  ( 2nd `  B
) x  =  ( y G z ) ) )
1716rexbidv 2415 . . . . . 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 2649 . . . . 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 3683 . . . 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 2132 . . 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 7139 . . . . . . 7  |-  Q.  e.  _V
2221a1i 9 . . . . . 6  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  Q.  e.  _V )
23 rabssab 3154 . . . . . . 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 7251 . . . . . . . . . . . 12  |-  ( f  e.  P.  ->  <. ( 1st `  f ) ,  ( 2nd `  f
) >.  e.  P. )
25 elprnql 7257 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  f
) ,  ( 2nd `  f ) >.  e.  P.  /\  y  e.  ( 1st `  f ) )  -> 
y  e.  Q. )
2624, 25sylan 281 . . . . . . . . . . 11  |-  ( ( f  e.  P.  /\  y  e.  ( 1st `  f ) )  -> 
y  e.  Q. )
27 prop 7251 . . . . . . . . . . . 12  |-  ( g  e.  P.  ->  <. ( 1st `  g ) ,  ( 2nd `  g
) >.  e.  P. )
28 elprnql 7257 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  g
) ,  ( 2nd `  g ) >.  e.  P.  /\  z  e.  ( 1st `  g ) )  -> 
z  e.  Q. )
2927, 28sylan 281 . . . . . . . . . . 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 2180 . . . . . . . . . . . 12  |-  ( x  =  ( y G z )  ->  (
x  e.  Q.  <->  ( y G z )  e. 
Q. ) )
3230, 31syl5ibrcom 156 . . . . . . . . . . 11  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( x  =  ( y G z )  ->  x  e.  Q. ) )
3326, 29, 32syl2an 287 . . . . . . . . . 10  |-  ( ( ( f  e.  P.  /\  y  e.  ( 1st `  f ) )  /\  ( g  e.  P.  /\  z  e.  ( 1st `  g ) ) )  ->  ( x  =  ( y G z )  ->  x  e.  Q. ) )
3433an4s 562 . . . . . . . . 9  |-  ( ( ( f  e.  P.  /\  g  e.  P. )  /\  ( y  e.  ( 1st `  f )  /\  z  e.  ( 1st `  g ) ) )  ->  (
x  =  ( y G z )  ->  x  e.  Q. )
)
3534rexlimdvva 2534 . . . . . . . 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 3141 . . . . . . 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 3078 . . . . . 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 4038 . . . . 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 3154 . . . . . . 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 7258 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  f
) ,  ( 2nd `  f ) >.  e.  P.  /\  y  e.  ( 2nd `  f ) )  -> 
y  e.  Q. )
4124, 40sylan 281 . . . . . . . . . . 11  |-  ( ( f  e.  P.  /\  y  e.  ( 2nd `  f ) )  -> 
y  e.  Q. )
42 elprnqu 7258 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  g
) ,  ( 2nd `  g ) >.  e.  P.  /\  z  e.  ( 2nd `  g ) )  -> 
z  e.  Q. )
4327, 42sylan 281 . . . . . . . . . . 11  |-  ( ( g  e.  P.  /\  z  e.  ( 2nd `  g ) )  -> 
z  e.  Q. )
4441, 43, 32syl2an 287 . . . . . . . . . 10  |-  ( ( ( f  e.  P.  /\  y  e.  ( 2nd `  f ) )  /\  ( g  e.  P.  /\  z  e.  ( 2nd `  g ) ) )  ->  ( x  =  ( y G z )  ->  x  e.  Q. ) )
4544an4s 562 . . . . . . . . 9  |-  ( ( ( f  e.  P.  /\  g  e.  P. )  /\  ( y  e.  ( 2nd `  f )  /\  z  e.  ( 2nd `  g ) ) )  ->  (
x  =  ( y G z )  ->  x  e.  Q. )
)
4645rexlimdvva 2534 . . . . . . . 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 3141 . . . . . . 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 3078 . . . . . 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 4038 . . . . 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 4539 . . . . 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 413 . . . 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 5389 . . . . . . . 8  |-  ( w  =  f  ->  ( 1st `  w )  =  ( 1st `  f
) )
5352rexeqdv 2610 . . . . . . 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 2649 . . . . . 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 5389 . . . . . . . 8  |-  ( w  =  f  ->  ( 2nd `  w )  =  ( 2nd `  f
) )
5655rexeqdv 2610 . . . . . . 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 2649 . . . . . 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 3683 . . . . 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 5389 . . . . . . . . 9  |-  ( v  =  g  ->  ( 1st `  v )  =  ( 1st `  g
) )
6059rexeqdv 2610 . . . . . . . 8  |-  ( v  =  g  ->  ( E. z  e.  ( 1st `  v ) x  =  ( y G z )  <->  E. z  e.  ( 1st `  g
) x  =  ( y G z ) ) )
6160rexbidv 2415 . . . . . . 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 2649 . . . . . 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 5389 . . . . . . . . 9  |-  ( v  =  g  ->  ( 2nd `  v )  =  ( 2nd `  g
) )
6463rexeqdv 2610 . . . . . . . 8  |-  ( v  =  g  ->  ( E. z  e.  ( 2nd `  v ) x  =  ( y G z )  <->  E. z  e.  ( 2nd `  g
) x  =  ( y G z ) ) )
6564rexbidv 2415 . . . . . . 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 2649 . . . . . 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 3683 . . . . 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 7284 . . . . 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 5873 . . . 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 1301 . . 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 2728 . 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 2124 . . . . . 6  |-  ( x  =  q  ->  (
x  =  ( y G z )  <->  q  =  ( y G z ) ) )
74732rexbidv 2437 . . . . 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 5749 . . . . . . 7  |-  ( y  =  r  ->  (
y G z )  =  ( r G z ) )
7675eqeq2d 2129 . . . . . 6  |-  ( y  =  r  ->  (
q  =  ( y G z )  <->  q  =  ( r G z ) ) )
77 oveq2 5750 . . . . . . 7  |-  ( z  =  s  ->  (
r G z )  =  ( r G s ) )
7877eqeq2d 2129 . . . . . 6  |-  ( z  =  s  ->  (
q  =  ( r G z )  <->  q  =  ( r G s ) ) )
7976, 78cbvrex2v 2640 . . . . 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, 79syl6bb 195 . . . 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 2659 . . 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 2437 . . . . 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 2640 . . . . 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, 83syl6bb 195 . . . 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 2659 . . 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 3680 . 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, 86syl6eq 2166 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 103    /\ w3a 947    = wceq 1316    e. wcel 1465   {cab 2103   E.wrex 2394   {crab 2397   _Vcvv 2660   <.cop 3500    X. cxp 4507   ` cfv 5093  (class class class)co 5742    e. cmpo 5744   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056   P.cnp 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-qs 6403  df-ni 7080  df-nqqs 7124  df-inp 7242
This theorem is referenced by:  genpelvl  7288  genpelvu  7289  plpvlu  7314  mpvlu  7315
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