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Theorem ltexprlemelu 6851
Description: Element in upper cut of the constructed difference. Lemma for ltexpri 6865. (Contributed by Jim Kingdon, 21-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
Assertion
Ref Expression
ltexprlemelu  |-  ( r  e.  ( 2nd `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
Distinct variable groups:    x, y, r, A    x, B, y, r    x, C, y, r

Proof of Theorem ltexprlemelu
StepHypRef Expression
1 oveq2 5551 . . . . 5  |-  ( x  =  r  ->  (
y  +Q  x )  =  ( y  +Q  r ) )
21eleq1d 2148 . . . 4  |-  ( x  =  r  ->  (
( y  +Q  x
)  e.  ( 2nd `  B )  <->  ( y  +Q  r )  e.  ( 2nd `  B ) ) )
32anbi2d 452 . . 3  |-  ( x  =  r  ->  (
( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) )  <->  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
43exbidv 1747 . 2  |-  ( x  =  r  ->  ( E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) )  <->  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
5 ltexprlem.1 . . . 4  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
65fveq2i 5212 . . 3  |-  ( 2nd `  C )  =  ( 2nd `  <. { x  e.  Q.  |  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >. )
7 nqex 6615 . . . . 5  |-  Q.  e.  _V
87rabex 3930 . . . 4  |-  { x  e.  Q.  |  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) }  e.  _V
97rabex 3930 . . . 4  |-  { x  e.  Q.  |  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }  e.  _V
108, 9op2nd 5805 . . 3  |-  ( 2nd `  <. { x  e. 
Q.  |  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >. )  =  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }
116, 10eqtri 2102 . 2  |-  ( 2nd `  C )  =  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }
124, 11elrab2 2752 1  |-  ( r  e.  ( 2nd `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   {crab 2353   <.cop 3409   ` cfv 4932  (class class class)co 5543   1stc1st 5796   2ndc2nd 5797   Q.cnq 6532    +Q cplq 6534
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-iinf 4337
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  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-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-2nd 5799  df-qs 6178  df-ni 6556  df-nqqs 6600
This theorem is referenced by:  ltexprlemm  6852  ltexprlemopu  6855  ltexprlemupu  6856  ltexprlemdisj  6858  ltexprlemloc  6859  ltexprlemfu  6863  ltexprlemru  6864
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