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Theorem recexprlemell 7542
Description: Membership in the lower cut of  B. Lemma for recexpr 7558. (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 2723 . 2  |-  ( C  e.  ( 1st `  B
)  ->  C  e.  _V )
2 ltrelnq 7285 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4638 . . . . . 6  |-  ( C 
<Q  y  ->  ( C  e.  Q.  /\  y  e.  Q. ) )
43simpld 111 . . . . 5  |-  ( C 
<Q  y  ->  C  e. 
Q. )
5 elex 2723 . . . . 5  |-  ( C  e.  Q.  ->  C  e.  _V )
64, 5syl 14 . . . 4  |-  ( C 
<Q  y  ->  C  e. 
_V )
76adantr 274 . . 3  |-  ( ( C  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  C  e.  _V )
87exlimiv 1578 . 2  |-  ( E. y ( C  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  C  e.  _V )
9 breq1 3968 . . . . 5  |-  ( x  =  C  ->  (
x  <Q  y  <->  C  <Q  y ) )
109anbi1d 461 . . . 4  |-  ( x  =  C  ->  (
( x  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  <->  ( C  <Q  y  /\  ( *Q
`  y )  e.  ( 2nd `  A
) ) ) )
1110exbidv 1805 . . 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 5471 . . . 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 7283 . . . . . 6  |-  Q.  e.  _V
152brel 4638 . . . . . . . . . 10  |-  ( x 
<Q  y  ->  ( x  e.  Q.  /\  y  e.  Q. ) )
1615simpld 111 . . . . . . . . 9  |-  ( x 
<Q  y  ->  x  e. 
Q. )
1716adantr 274 . . . . . . . 8  |-  ( ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  x  e.  Q. )
1817exlimiv 1578 . . . . . . 7  |-  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
1918abssi 3203 . . . . . 6  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q.
2014, 19ssexi 4102 . . . . 5  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  e.  _V
212brel 4638 . . . . . . . . . 10  |-  ( y 
<Q  x  ->  ( y  e.  Q.  /\  x  e.  Q. ) )
2221simprd 113 . . . . . . . . 9  |-  ( y 
<Q  x  ->  x  e. 
Q. )
2322adantr 274 . . . . . . . 8  |-  ( ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  x  e.  Q. )
2423exlimiv 1578 . . . . . . 7  |-  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  x  e.  Q. )
2524abssi 3203 . . . . . 6  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q.
2614, 25ssexi 4102 . . . . 5  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  e.  _V
2720, 26op1st 6094 . . . 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 2178 . . 3  |-  ( 1st `  B )  =  {
x  |  E. y
( x  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) }
2911, 28elab2g 2859 . 2  |-  ( C  e.  _V  ->  ( C  e.  ( 1st `  B )  <->  E. y
( C  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) ) )
301, 8, 29pm5.21nii 694 1  |-  ( C  e.  ( 1st `  B
)  <->  E. y ( C 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1335   E.wex 1472    e. wcel 2128   {cab 2143   _Vcvv 2712   <.cop 3563   class class class wbr 3965   ` cfv 5170   1stc1st 6086   2ndc2nd 6087   Q.cnq 7200   *Qcrq 7204    <Q cltq 7205
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-1st 6088  df-qs 6486  df-ni 7224  df-nqqs 7268  df-ltnqqs 7273
This theorem is referenced by:  recexprlemm  7544  recexprlemopl  7545  recexprlemlol  7546  recexprlemdisj  7550  recexprlemloc  7551  recexprlem1ssl  7553  recexprlemss1l  7555
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