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Theorem recexprlemelu 7622
Description: Membership in the upper cut of  B. Lemma for recexpr 7637. (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 2749 . 2  |-  ( C  e.  ( 2nd `  B
)  ->  C  e.  _V )
2 ltrelnq 7364 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4679 . . . . . 6  |-  ( y 
<Q  C  ->  ( y  e.  Q.  /\  C  e.  Q. ) )
43simprd 114 . . . . 5  |-  ( y 
<Q  C  ->  C  e. 
Q. )
5 elex 2749 . . . . 5  |-  ( C  e.  Q.  ->  C  e.  _V )
64, 5syl 14 . . . 4  |-  ( y 
<Q  C  ->  C  e. 
_V )
76adantr 276 . . 3  |-  ( ( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  C  e.  _V )
87exlimiv 1598 . 2  |-  ( E. y ( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  C  e.  _V )
9 breq2 4008 . . . . 5  |-  ( x  =  C  ->  (
y  <Q  x  <->  y  <Q  C ) )
109anbi1d 465 . . . 4  |-  ( x  =  C  ->  (
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( y  <Q  C  /\  ( *Q
`  y )  e.  ( 1st `  A
) ) ) )
1110exbidv 1825 . . 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 5519 . . . 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 7362 . . . . . 6  |-  Q.  e.  _V
152brel 4679 . . . . . . . . . 10  |-  ( x 
<Q  y  ->  ( x  e.  Q.  /\  y  e.  Q. ) )
1615simpld 112 . . . . . . . . 9  |-  ( x 
<Q  y  ->  x  e. 
Q. )
1716adantr 276 . . . . . . . 8  |-  ( ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  x  e.  Q. )
1817exlimiv 1598 . . . . . . 7  |-  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
1918abssi 3231 . . . . . 6  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q.
2014, 19ssexi 4142 . . . . 5  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  e.  _V
212brel 4679 . . . . . . . . . 10  |-  ( y 
<Q  x  ->  ( y  e.  Q.  /\  x  e.  Q. ) )
2221simprd 114 . . . . . . . . 9  |-  ( y 
<Q  x  ->  x  e. 
Q. )
2322adantr 276 . . . . . . . 8  |-  ( ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  x  e.  Q. )
2423exlimiv 1598 . . . . . . 7  |-  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  x  e.  Q. )
2524abssi 3231 . . . . . 6  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q.
2614, 25ssexi 4142 . . . . 5  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  e.  _V
2720, 26op2nd 6148 . . . 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 2198 . . 3  |-  ( 2nd `  B )  =  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
2911, 28elab2g 2885 . 2  |-  ( C  e.  _V  ->  ( C  e.  ( 2nd `  B )  <->  E. y
( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) ) )
301, 8, 29pm5.21nii 704 1  |-  ( C  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   _Vcvv 2738   <.cop 3596   class class class wbr 4004   ` cfv 5217   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279   *Qcrq 7283    <Q cltq 7284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-2nd 6142  df-qs 6541  df-ni 7303  df-nqqs 7347  df-ltnqqs 7352
This theorem is referenced by:  recexprlemm  7623  recexprlemopu  7626  recexprlemupu  7627  recexprlemdisj  7629  recexprlemloc  7630  recexprlem1ssu  7633  recexprlemss1u  7635
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