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Theorem ltdfpr 7502
Description: More convenient form of df-iltp 7466. (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 4003 . . 3  |-  ( A 
<P  B  <->  <. A ,  B >.  e.  <P  )
2 df-iltp 7466 . . . 4  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }
32eleq2i 2244 . . 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 184 . 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 109 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
65fveq2d 5518 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 2nd `  x
)  =  ( 2nd `  A ) )
76eleq2d 2247 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( q  e.  ( 2nd `  x )  <-> 
q  e.  ( 2nd `  A ) ) )
8 simpr 110 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
98fveq2d 5518 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 1st `  y
)  =  ( 1st `  B ) )
109eleq2d 2247 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( q  e.  ( 1st `  y )  <-> 
q  e.  ( 1st `  B ) ) )
117, 10anbi12d 473 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( q  e.  ( 2nd `  x
)  /\  q  e.  ( 1st `  y ) )  <->  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  B ) ) ) )
1211rexbidv 2478 . . 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 4264 . 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, 13bitrid 192 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   E.wrex 2456   <.cop 3595   class class class wbr 4002   {copab 4062   ` cfv 5215   1stc1st 6136   2ndc2nd 6137   Q.cnq 7276   P.cnp 7287    <P cltp 7291
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-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-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
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-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-iota 5177  df-fv 5223  df-iltp 7466
This theorem is referenced by:  nqprl  7547  nqpru  7548  ltprordil  7585  ltnqpr  7589  ltnqpri  7590  ltpopr  7591  ltsopr  7592  ltaddpr  7593  ltexprlemm  7596  ltexprlemopu  7599  ltexprlemru  7608  aptiprleml  7635  aptiprlemu  7636  archpr  7639  cauappcvgprlem2  7656  caucvgprlem2  7676  caucvgprprlemopu  7695  caucvgprprlemexbt  7702  caucvgprprlem2  7706  suplocexprlemloc  7717  suplocexprlemub  7719  suplocexprlemlub  7720
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