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Theorem recexprlemell 7584
Description: Membership in the lower cut of  B. Lemma for recexpr 7600. (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 2741 . 2  |-  ( C  e.  ( 1st `  B
)  ->  C  e.  _V )
2 ltrelnq 7327 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4663 . . . . . 6  |-  ( C 
<Q  y  ->  ( C  e.  Q.  /\  y  e.  Q. ) )
43simpld 111 . . . . 5  |-  ( C 
<Q  y  ->  C  e. 
Q. )
5 elex 2741 . . . . 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 1591 . 2  |-  ( E. y ( C  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  C  e.  _V )
9 breq1 3992 . . . . 5  |-  ( x  =  C  ->  (
x  <Q  y  <->  C  <Q  y ) )
109anbi1d 462 . . . 4  |-  ( x  =  C  ->  (
( x  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  <->  ( C  <Q  y  /\  ( *Q
`  y )  e.  ( 2nd `  A
) ) ) )
1110exbidv 1818 . . 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 5499 . . . 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 7325 . . . . . 6  |-  Q.  e.  _V
152brel 4663 . . . . . . . . . 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 1591 . . . . . . 7  |-  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
1918abssi 3222 . . . . . 6  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q.
2014, 19ssexi 4127 . . . . 5  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  e.  _V
212brel 4663 . . . . . . . . . 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 1591 . . . . . . 7  |-  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  x  e.  Q. )
2524abssi 3222 . . . . . 6  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q.
2614, 25ssexi 4127 . . . . 5  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  e.  _V
2720, 26op1st 6125 . . . 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 2191 . . 3  |-  ( 1st `  B )  =  {
x  |  E. y
( x  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) }
2911, 28elab2g 2877 . 2  |-  ( C  e.  _V  ->  ( C  e.  ( 1st `  B )  <->  E. y
( C  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) ) )
301, 8, 29pm5.21nii 699 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 1348   E.wex 1485    e. wcel 2141   {cab 2156   _Vcvv 2730   <.cop 3586   class class class wbr 3989   ` cfv 5198   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   *Qcrq 7246    <Q cltq 7247
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-qs 6519  df-ni 7266  df-nqqs 7310  df-ltnqqs 7315
This theorem is referenced by:  recexprlemm  7586  recexprlemopl  7587  recexprlemlol  7588  recexprlemdisj  7592  recexprlemloc  7593  recexprlem1ssl  7595  recexprlemss1l  7597
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