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Theorem recexprlemell 6778
Description: Membership in the lower cut of  B. Lemma for recexpr 6794. (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
recexprlemell  |-  ( C  e.  ( 1st `  B
)  <->  E. y ( C 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y

Proof of Theorem recexprlemell
StepHypRef Expression
1 elex 2583 . 2  |-  ( C  e.  ( 1st `  B
)  ->  C  e.  _V )
2 ltrelnq 6521 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4420 . . . . . 6  |-  ( C 
<Q  y  ->  ( C  e.  Q.  /\  y  e.  Q. ) )
43simpld 109 . . . . 5  |-  ( C 
<Q  y  ->  C  e. 
Q. )
5 elex 2583 . . . . 5  |-  ( C  e.  Q.  ->  C  e.  _V )
64, 5syl 14 . . . 4  |-  ( C 
<Q  y  ->  C  e. 
_V )
76adantr 265 . . 3  |-  ( ( C  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  C  e.  _V )
87exlimiv 1505 . 2  |-  ( E. y ( C  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  C  e.  _V )
9 breq1 3795 . . . . 5  |-  ( x  =  C  ->  (
x  <Q  y  <->  C  <Q  y ) )
109anbi1d 446 . . . 4  |-  ( x  =  C  ->  (
( x  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  <->  ( C  <Q  y  /\  ( *Q
`  y )  e.  ( 2nd `  A
) ) ) )
1110exbidv 1722 . . 3  |-  ( x  =  C  ->  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  <->  E. y ( C 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  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 5209 . . . 4  |-  ( 1st `  B )  =  ( 1st `  <. { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) } ,  { x  |  E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >. )
14 nqex 6519 . . . . . 6  |-  Q.  e.  _V
152brel 4420 . . . . . . . . . 10  |-  ( x 
<Q  y  ->  ( x  e.  Q.  /\  y  e.  Q. ) )
1615simpld 109 . . . . . . . . 9  |-  ( x 
<Q  y  ->  x  e. 
Q. )
1716adantr 265 . . . . . . . 8  |-  ( ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  x  e.  Q. )
1817exlimiv 1505 . . . . . . 7  |-  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
1918abssi 3043 . . . . . 6  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q.
2014, 19ssexi 3923 . . . . 5  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  e.  _V
212brel 4420 . . . . . . . . . 10  |-  ( y 
<Q  x  ->  ( y  e.  Q.  /\  x  e.  Q. ) )
2221simprd 111 . . . . . . . . 9  |-  ( y 
<Q  x  ->  x  e. 
Q. )
2322adantr 265 . . . . . . . 8  |-  ( ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  x  e.  Q. )
2423exlimiv 1505 . . . . . . 7  |-  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  x  e.  Q. )
2524abssi 3043 . . . . . 6  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q.
2614, 25ssexi 3923 . . . . 5  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  e.  _V
2720, 26op1st 5801 . . . 4  |-  ( 1st `  <. { 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 ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) }
2813, 27eqtri 2076 . . 3  |-  ( 1st `  B )  =  {
x  |  E. y
( x  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) }
2911, 28elab2g 2712 . 2  |-  ( C  e.  _V  ->  ( C  e.  ( 1st `  B )  <->  E. y
( C  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) ) )
301, 8, 29pm5.21nii 630 1  |-  ( C  e.  ( 1st `  B
)  <->  E. y ( C 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   _Vcvv 2574   <.cop 3406   class class class wbr 3792   ` cfv 4930   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436   *Qcrq 6440    <Q cltq 6441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-1st 5795  df-qs 6143  df-ni 6460  df-nqqs 6504  df-ltnqqs 6509
This theorem is referenced by:  recexprlemm  6780  recexprlemopl  6781  recexprlemlol  6782  recexprlemdisj  6786  recexprlemloc  6787  recexprlem1ssl  6789  recexprlemss1l  6791
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