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Theorem ltdfpr 6661
Description: More convenient form of df-iltp 6625. (Contributed by Jim Kingdon, 15-Dec-2019.)
Assertion
Ref Expression
ltdfpr  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  B ) ) ) )
Distinct variable groups:    A, q    B, q

Proof of Theorem ltdfpr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 3792 . . 3  |-  ( A 
<P  B  <->  <. A ,  B >.  e.  <P  )
2 df-iltp 6625 . . . 4  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }
32eleq2i 2120 . . 3  |-  ( <. A ,  B >.  e. 
<P 
<-> 
<. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) } )
41, 3bitri 177 . 2  |-  ( A 
<P  B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) } )
5 simpl 106 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
65fveq2d 5209 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 2nd `  x
)  =  ( 2nd `  A ) )
76eleq2d 2123 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( q  e.  ( 2nd `  x )  <-> 
q  e.  ( 2nd `  A ) ) )
8 simpr 107 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
98fveq2d 5209 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 1st `  y
)  =  ( 1st `  B ) )
109eleq2d 2123 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( q  e.  ( 1st `  y )  <-> 
q  e.  ( 1st `  B ) ) )
117, 10anbi12d 450 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( q  e.  ( 2nd `  x
)  /\  q  e.  ( 1st `  y ) )  <->  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  B ) ) ) )
1211rexbidv 2344 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( E. q  e. 
Q.  ( q  e.  ( 2nd `  x
)  /\  q  e.  ( 1st `  y ) )  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  B ) ) ) )
1312opelopab2a 4029 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  B ) ) ) )
144, 13syl5bb 185 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   E.wrex 2324   <.cop 3405   class class class wbr 3791   {copab 3844   ` cfv 4929   1stc1st 5792   2ndc2nd 5793   Q.cnq 6435   P.cnp 6446    <P cltp 6450
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-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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
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-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-iota 4894  df-fv 4937  df-iltp 6625
This theorem is referenced by:  nqprl  6706  nqpru  6707  ltprordil  6744  ltnqpr  6748  ltnqpri  6749  ltpopr  6750  ltsopr  6751  ltaddpr  6752  ltexprlemm  6755  ltexprlemopu  6758  ltexprlemru  6767  aptiprleml  6794  aptiprlemu  6795  archpr  6798  cauappcvgprlem2  6815  caucvgprlem2  6835  caucvgprprlemopu  6854  caucvgprprlemexbt  6861  caucvgprprlem2  6865
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