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Theorem ltdfpr 7278
Description: More convenient form of df-iltp 7242. (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 3898 . . 3  |-  ( A 
<P  B  <->  <. A ,  B >.  e.  <P  )
2 df-iltp 7242 . . . 4  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }
32eleq2i 2182 . . 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 183 . 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 108 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
65fveq2d 5391 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 2nd `  x
)  =  ( 2nd `  A ) )
76eleq2d 2185 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( q  e.  ( 2nd `  x )  <-> 
q  e.  ( 2nd `  A ) ) )
8 simpr 109 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
98fveq2d 5391 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 1st `  y
)  =  ( 1st `  B ) )
109eleq2d 2185 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( q  e.  ( 1st `  y )  <-> 
q  e.  ( 1st `  B ) ) )
117, 10anbi12d 462 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( q  e.  ( 2nd `  x
)  /\  q  e.  ( 1st `  y ) )  <->  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  B ) ) ) )
1211rexbidv 2413 . . 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 4155 . 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 191 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 103    <-> wb 104    = wceq 1314    e. wcel 1463   E.wrex 2392   <.cop 3498   class class class wbr 3897   {copab 3956   ` cfv 5091   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052   P.cnp 7063    <P cltp 7067
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-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-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
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-rex 2397  df-v 2660  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-br 3898  df-opab 3958  df-iota 5056  df-fv 5099  df-iltp 7242
This theorem is referenced by:  nqprl  7323  nqpru  7324  ltprordil  7361  ltnqpr  7365  ltnqpri  7366  ltpopr  7367  ltsopr  7368  ltaddpr  7369  ltexprlemm  7372  ltexprlemopu  7375  ltexprlemru  7384  aptiprleml  7411  aptiprlemu  7412  archpr  7415  cauappcvgprlem2  7432  caucvgprlem2  7452  caucvgprprlemopu  7471  caucvgprprlemexbt  7478  caucvgprprlem2  7482  suplocexprlemloc  7493  suplocexprlemub  7495  suplocexprlemlub  7496
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