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Theorem ltdfpr 7282
Description: More convenient form of df-iltp 7246. (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 3900 . . 3  |-  ( A 
<P  B  <->  <. A ,  B >.  e.  <P  )
2 df-iltp 7246 . . . 4  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }
32eleq2i 2184 . . 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 5393 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 2nd `  x
)  =  ( 2nd `  A ) )
76eleq2d 2187 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( q  e.  ( 2nd `  x )  <-> 
q  e.  ( 2nd `  A ) ) )
8 simpr 109 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
98fveq2d 5393 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 1st `  y
)  =  ( 1st `  B ) )
109eleq2d 2187 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( q  e.  ( 1st `  y )  <-> 
q  e.  ( 1st `  B ) ) )
117, 10anbi12d 464 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( q  e.  ( 2nd `  x
)  /\  q  e.  ( 1st `  y ) )  <->  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  B ) ) ) )
1211rexbidv 2415 . . 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 4157 . 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 1316    e. wcel 1465   E.wrex 2394   <.cop 3500   class class class wbr 3899   {copab 3958   ` cfv 5093   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056   P.cnp 7067    <P cltp 7071
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-iota 5058  df-fv 5101  df-iltp 7246
This theorem is referenced by:  nqprl  7327  nqpru  7328  ltprordil  7365  ltnqpr  7369  ltnqpri  7370  ltpopr  7371  ltsopr  7372  ltaddpr  7373  ltexprlemm  7376  ltexprlemopu  7379  ltexprlemru  7388  aptiprleml  7415  aptiprlemu  7416  archpr  7419  cauappcvgprlem2  7436  caucvgprlem2  7456  caucvgprprlemopu  7475  caucvgprprlemexbt  7482  caucvgprprlem2  7486  suplocexprlemloc  7497  suplocexprlemub  7499  suplocexprlemlub  7500
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