ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recexprlemelu Unicode version

Theorem recexprlemelu 7395
Description: Membership in the upper cut of  B. Lemma for recexpr 7410. (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 2669 . 2  |-  ( C  e.  ( 2nd `  B
)  ->  C  e.  _V )
2 ltrelnq 7137 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4559 . . . . . 6  |-  ( y 
<Q  C  ->  ( y  e.  Q.  /\  C  e.  Q. ) )
43simprd 113 . . . . 5  |-  ( y 
<Q  C  ->  C  e. 
Q. )
5 elex 2669 . . . . 5  |-  ( C  e.  Q.  ->  C  e.  _V )
64, 5syl 14 . . . 4  |-  ( y 
<Q  C  ->  C  e. 
_V )
76adantr 272 . . 3  |-  ( ( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  C  e.  _V )
87exlimiv 1560 . 2  |-  ( E. y ( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  C  e.  _V )
9 breq2 3901 . . . . 5  |-  ( x  =  C  ->  (
y  <Q  x  <->  y  <Q  C ) )
109anbi1d 458 . . . 4  |-  ( x  =  C  ->  (
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( y  <Q  C  /\  ( *Q
`  y )  e.  ( 1st `  A
) ) ) )
1110exbidv 1779 . . 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 5390 . . . 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 7135 . . . . . 6  |-  Q.  e.  _V
152brel 4559 . . . . . . . . . 10  |-  ( x 
<Q  y  ->  ( x  e.  Q.  /\  y  e.  Q. ) )
1615simpld 111 . . . . . . . . 9  |-  ( x 
<Q  y  ->  x  e. 
Q. )
1716adantr 272 . . . . . . . 8  |-  ( ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  x  e.  Q. )
1817exlimiv 1560 . . . . . . 7  |-  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
1918abssi 3140 . . . . . 6  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q.
2014, 19ssexi 4034 . . . . 5  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  e.  _V
212brel 4559 . . . . . . . . . 10  |-  ( y 
<Q  x  ->  ( y  e.  Q.  /\  x  e.  Q. ) )
2221simprd 113 . . . . . . . . 9  |-  ( y 
<Q  x  ->  x  e. 
Q. )
2322adantr 272 . . . . . . . 8  |-  ( ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  x  e.  Q. )
2423exlimiv 1560 . . . . . . 7  |-  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  x  e.  Q. )
2524abssi 3140 . . . . . 6  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q.
2614, 25ssexi 4034 . . . . 5  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  e.  _V
2720, 26op2nd 6011 . . . 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 2136 . . 3  |-  ( 2nd `  B )  =  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
2911, 28elab2g 2802 . 2  |-  ( C  e.  _V  ->  ( C  e.  ( 2nd `  B )  <->  E. y
( y  <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) ) )
301, 8, 29pm5.21nii 676 1  |-  ( C  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  C  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   _Vcvv 2658   <.cop 3498   class class class wbr 3897   ` cfv 5091   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052   *Qcrq 7056    <Q cltq 7057
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-2nd 6005  df-qs 6401  df-ni 7076  df-nqqs 7120  df-ltnqqs 7125
This theorem is referenced by:  recexprlemm  7396  recexprlemopu  7399  recexprlemupu  7400  recexprlemdisj  7402  recexprlemloc  7403  recexprlem1ssu  7406  recexprlemss1u  7408
  Copyright terms: Public domain W3C validator