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Theorem genipv 6761
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 5550 . . . 4  |-  ( f  =  A  ->  (
f F g )  =  ( A F g ) )
2 fveq2 5209 . . . . . . 7  |-  ( f  =  A  ->  ( 1st `  f )  =  ( 1st `  A
) )
32rexeqdv 2557 . . . . . 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 2594 . . . . 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 5209 . . . . . . 7  |-  ( f  =  A  ->  ( 2nd `  f )  =  ( 2nd `  A
) )
65rexeqdv 2557 . . . . . 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 2594 . . . . 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 3586 . . . 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 2096 . . 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 5551 . . . 4  |-  ( g  =  B  ->  ( A F g )  =  ( A F B ) )
11 fveq2 5209 . . . . . . . 8  |-  ( g  =  B  ->  ( 1st `  g )  =  ( 1st `  B
) )
1211rexeqdv 2557 . . . . . . 7  |-  ( g  =  B  ->  ( E. z  e.  ( 1st `  g ) x  =  ( y G z )  <->  E. z  e.  ( 1st `  B
) x  =  ( y G z ) ) )
1312rexbidv 2370 . . . . . 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 2594 . . . . 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 5209 . . . . . . . 8  |-  ( g  =  B  ->  ( 2nd `  g )  =  ( 2nd `  B
) )
1615rexeqdv 2557 . . . . . . 7  |-  ( g  =  B  ->  ( E. z  e.  ( 2nd `  g ) x  =  ( y G z )  <->  E. z  e.  ( 2nd `  B
) x  =  ( y G z ) ) )
1716rexbidv 2370 . . . . . 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 2594 . . . . 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 3586 . . . 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 2096 . . 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 6615 . . . . . . 7  |-  Q.  e.  _V
2221a1i 9 . . . . . 6  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  Q.  e.  _V )
23 rabssab 3082 . . . . . . 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 6727 . . . . . . . . . . . 12  |-  ( f  e.  P.  ->  <. ( 1st `  f ) ,  ( 2nd `  f
) >.  e.  P. )
25 elprnql 6733 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  f
) ,  ( 2nd `  f ) >.  e.  P.  /\  y  e.  ( 1st `  f ) )  -> 
y  e.  Q. )
2624, 25sylan 277 . . . . . . . . . . 11  |-  ( ( f  e.  P.  /\  y  e.  ( 1st `  f ) )  -> 
y  e.  Q. )
27 prop 6727 . . . . . . . . . . . 12  |-  ( g  e.  P.  ->  <. ( 1st `  g ) ,  ( 2nd `  g
) >.  e.  P. )
28 elprnql 6733 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  g
) ,  ( 2nd `  g ) >.  e.  P.  /\  z  e.  ( 1st `  g ) )  -> 
z  e.  Q. )
2927, 28sylan 277 . . . . . . . . . . 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 2142 . . . . . . . . . . . 12  |-  ( x  =  ( y G z )  ->  (
x  e.  Q.  <->  ( y G z )  e. 
Q. ) )
3230, 31syl5ibrcom 155 . . . . . . . . . . 11  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( x  =  ( y G z )  ->  x  e.  Q. ) )
3326, 29, 32syl2an 283 . . . . . . . . . 10  |-  ( ( ( f  e.  P.  /\  y  e.  ( 1st `  f ) )  /\  ( g  e.  P.  /\  z  e.  ( 1st `  g ) ) )  ->  ( x  =  ( y G z )  ->  x  e.  Q. ) )
3433an4s 553 . . . . . . . . 9  |-  ( ( ( f  e.  P.  /\  g  e.  P. )  /\  ( y  e.  ( 1st `  f )  /\  z  e.  ( 1st `  g ) ) )  ->  (
x  =  ( y G z )  ->  x  e.  Q. )
)
3534rexlimdvva 2485 . . . . . . . 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 3069 . . . . . . 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, 36syl5ss 3011 . . . . . 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 3926 . . . . 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 3082 . . . . . . 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 6734 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  f
) ,  ( 2nd `  f ) >.  e.  P.  /\  y  e.  ( 2nd `  f ) )  -> 
y  e.  Q. )
4124, 40sylan 277 . . . . . . . . . . 11  |-  ( ( f  e.  P.  /\  y  e.  ( 2nd `  f ) )  -> 
y  e.  Q. )
42 elprnqu 6734 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  g
) ,  ( 2nd `  g ) >.  e.  P.  /\  z  e.  ( 2nd `  g ) )  -> 
z  e.  Q. )
4327, 42sylan 277 . . . . . . . . . . 11  |-  ( ( g  e.  P.  /\  z  e.  ( 2nd `  g ) )  -> 
z  e.  Q. )
4441, 43, 32syl2an 283 . . . . . . . . . 10  |-  ( ( ( f  e.  P.  /\  y  e.  ( 2nd `  f ) )  /\  ( g  e.  P.  /\  z  e.  ( 2nd `  g ) ) )  ->  ( x  =  ( y G z )  ->  x  e.  Q. ) )
4544an4s 553 . . . . . . . . 9  |-  ( ( ( f  e.  P.  /\  g  e.  P. )  /\  ( y  e.  ( 2nd `  f )  /\  z  e.  ( 2nd `  g ) ) )  ->  (
x  =  ( y G z )  ->  x  e.  Q. )
)
4645rexlimdvva 2485 . . . . . . . 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 3069 . . . . . . 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, 47syl5ss 3011 . . . . . 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 3926 . . . . 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 4400 . . . . 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 408 . . . 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 5209 . . . . . . . 8  |-  ( w  =  f  ->  ( 1st `  w )  =  ( 1st `  f
) )
5352rexeqdv 2557 . . . . . . 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 2594 . . . . . 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 5209 . . . . . . . 8  |-  ( w  =  f  ->  ( 2nd `  w )  =  ( 2nd `  f
) )
5655rexeqdv 2557 . . . . . . 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 2594 . . . . . 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 3586 . . . . 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 5209 . . . . . . . . 9  |-  ( v  =  g  ->  ( 1st `  v )  =  ( 1st `  g
) )
6059rexeqdv 2557 . . . . . . . 8  |-  ( v  =  g  ->  ( E. z  e.  ( 1st `  v ) x  =  ( y G z )  <->  E. z  e.  ( 1st `  g
) x  =  ( y G z ) ) )
6160rexbidv 2370 . . . . . . 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 2594 . . . . . 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 5209 . . . . . . . . 9  |-  ( v  =  g  ->  ( 2nd `  v )  =  ( 2nd `  g
) )
6463rexeqdv 2557 . . . . . . . 8  |-  ( v  =  g  ->  ( E. z  e.  ( 2nd `  v ) x  =  ( y G z )  <->  E. z  e.  ( 2nd `  g
) x  =  ( y G z ) ) )
6564rexbidv 2370 . . . . . . 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 2594 . . . . . 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 3586 . . . . 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 6760 . . . . 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, 69ovmpt2g 5666 . . . 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 1270 . . 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 2667 . 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 2088 . . . . . 6  |-  ( x  =  q  ->  (
x  =  ( y G z )  <->  q  =  ( y G z ) ) )
74732rexbidv 2392 . . . . 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 5550 . . . . . . 7  |-  ( y  =  r  ->  (
y G z )  =  ( r G z ) )
7675eqeq2d 2093 . . . . . 6  |-  ( y  =  r  ->  (
q  =  ( y G z )  <->  q  =  ( r G z ) ) )
77 oveq2 5551 . . . . . . 7  |-  ( z  =  s  ->  (
r G z )  =  ( r G s ) )
7877eqeq2d 2093 . . . . . 6  |-  ( z  =  s  ->  (
q  =  ( r G z )  <->  q  =  ( r G s ) ) )
7976, 78cbvrex2v 2587 . . . . 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 194 . . . 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 2601 . . 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 2392 . . . . 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 2587 . . . . 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 194 . . . 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 2601 . . 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 3583 . 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 2130 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 102    /\ w3a 920    = wceq 1285    e. wcel 1434   {cab 2068   E.wrex 2350   {crab 2353   _Vcvv 2602   <.cop 3409    X. cxp 4369   ` cfv 4932  (class class class)co 5543    |-> cmpt2 5545   1stc1st 5796   2ndc2nd 5797   Q.cnq 6532   P.cnp 6543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-qs 6178  df-ni 6556  df-nqqs 6600  df-inp 6718
This theorem is referenced by:  genpelvl  6764  genpelvu  6765  plpvlu  6790  mpvlu  6791
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