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Theorem ltdfpr 7505
Description: More convenient form of df-iltp 7469. (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 4005 . . 3  |-  ( A 
<P  B  <->  <. A ,  B >.  e.  <P  )
2 df-iltp 7469 . . . 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 5520 . . . . . 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 5520 . . . . . 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 4266 . 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 3596   class class class wbr 4004   {copab 4064   ` cfv 5217   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279   P.cnp 7290    <P cltp 7294
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 4122  ax-pow 4175  ax-pr 4210
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 2740  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-br 4005  df-opab 4066  df-iota 5179  df-fv 5225  df-iltp 7469
This theorem is referenced by:  nqprl  7550  nqpru  7551  ltprordil  7588  ltnqpr  7592  ltnqpri  7593  ltpopr  7594  ltsopr  7595  ltaddpr  7596  ltexprlemm  7599  ltexprlemopu  7602  ltexprlemru  7611  aptiprleml  7638  aptiprlemu  7639  archpr  7642  cauappcvgprlem2  7659  caucvgprlem2  7679  caucvgprprlemopu  7698  caucvgprprlemexbt  7705  caucvgprprlem2  7709  suplocexprlemloc  7720  suplocexprlemub  7722  suplocexprlemlub  7723
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