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Theorem recexprlemelu 7161
Description: Membership in the upper cut of  B. Lemma for recexpr 7176. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
Assertion
Ref Expression
recexprlemelu  |-  ( C  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y

Proof of Theorem recexprlemelu
StepHypRef Expression
1 elex 2630 . 2  |-  ( C  e.  ( 2nd `  B
)  ->  C  e.  _V )
2 ltrelnq 6903 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4478 . . . . . 6  |-  ( y 
<Q  C  ->  ( y  e.  Q.  /\  C  e.  Q. ) )
43simprd 112 . . . . 5  |-  ( y 
<Q  C  ->  C  e. 
Q. )
5 elex 2630 . . . . 5  |-  ( C  e.  Q.  ->  C  e.  _V )
64, 5syl 14 . . . 4  |-  ( y 
<Q  C  ->  C  e. 
_V )
76adantr 270 . . 3  |-  ( ( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  C  e.  _V )
87exlimiv 1534 . 2  |-  ( E. y ( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  C  e.  _V )
9 breq2 3841 . . . . 5  |-  ( x  =  C  ->  (
y  <Q  x  <->  y  <Q  C ) )
109anbi1d 453 . . . 4  |-  ( x  =  C  ->  (
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( y  <Q  C  /\  ( *Q
`  y )  e.  ( 1st `  A
) ) ) )
1110exbidv 1753 . . 3  |-  ( x  =  C  ->  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  <->  E. y ( y 
<Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) ) )
12 recexpr.1 . . . . 5  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
1312fveq2i 5292 . . . 4  |-  ( 2nd `  B )  =  ( 2nd `  <. { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) } ,  { x  |  E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >. )
14 nqex 6901 . . . . . 6  |-  Q.  e.  _V
152brel 4478 . . . . . . . . . 10  |-  ( x 
<Q  y  ->  ( x  e.  Q.  /\  y  e.  Q. ) )
1615simpld 110 . . . . . . . . 9  |-  ( x 
<Q  y  ->  x  e. 
Q. )
1716adantr 270 . . . . . . . 8  |-  ( ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  x  e.  Q. )
1817exlimiv 1534 . . . . . . 7  |-  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
1918abssi 3094 . . . . . 6  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q.
2014, 19ssexi 3969 . . . . 5  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  e.  _V
212brel 4478 . . . . . . . . . 10  |-  ( y 
<Q  x  ->  ( y  e.  Q.  /\  x  e.  Q. ) )
2221simprd 112 . . . . . . . . 9  |-  ( y 
<Q  x  ->  x  e. 
Q. )
2322adantr 270 . . . . . . . 8  |-  ( ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  x  e.  Q. )
2423exlimiv 1534 . . . . . . 7  |-  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  x  e.  Q. )
2524abssi 3094 . . . . . 6  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q.
2614, 25ssexi 3969 . . . . 5  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  e.  _V
2720, 26op2nd 5900 . . . 4  |-  ( 2nd `  <. { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) } ,  { x  |  E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >. )  =  { x  |  E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) }
2813, 27eqtri 2108 . . 3  |-  ( 2nd `  B )  =  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
2911, 28elab2g 2760 . 2  |-  ( C  e.  _V  ->  ( C  e.  ( 2nd `  B )  <->  E. y
( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) ) )
301, 8, 29pm5.21nii 655 1  |-  ( C  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   _Vcvv 2619   <.cop 3444   class class class wbr 3837   ` cfv 5002   1stc1st 5891   2ndc2nd 5892   Q.cnq 6818   *Qcrq 6822    <Q cltq 6823
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-2nd 5894  df-qs 6278  df-ni 6842  df-nqqs 6886  df-ltnqqs 6891
This theorem is referenced by:  recexprlemm  7162  recexprlemopu  7165  recexprlemupu  7166  recexprlemdisj  7168  recexprlemloc  7169  recexprlem1ssu  7172  recexprlemss1u  7174
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