Theorem ltdfpr 7621

Description: More convenient form of df-iltp 7585. (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.

Step	Hyp	Ref	Expression
1		df-br 4046	. . 3 |- ( A <P B <-> <. A , B >. e. <P )
2		df-iltp 7585	. . . 4 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
3	2	eleq2i 2272	. . 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 ) ) ) } )
4	1, 3	bitri 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 )
6	5	fveq2d 5582	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( 2nd ` x ) = ( 2nd ` A ) )
7	6	eleq2d 2275	. . . . 5 |- ( ( x = A /\ y = B ) -> ( q e. ( 2nd ` x ) <-> q e. ( 2nd ` A ) ) )
8		simpr 110	. . . . . . 7 |- ( ( x = A /\ y = B ) -> y = B )
9	8	fveq2d 5582	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( 1st ` y ) = ( 1st ` B ) )
10	9	eleq2d 2275	. . . . 5 |- ( ( x = A /\ y = B ) -> ( q e. ( 1st ` y ) <-> q e. ( 1st ` B ) ) )
11	7, 10	anbi12d 473	. . . 4 |- ( ( x = A /\ y = B ) -> ( ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) <-> ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) )
12	11	rexbidv 2507	. . 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 ) ) ) )
13	12	opelopab2a 4312	. 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 ) ) ) )
14	4, 13	bitrid 192	1 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   E.wrex 2485   <.cop 3636   class class class wbr 4045   {copab 4105   ` cfv 5272   1stc1st 6226   2ndc2nd 6227   Q.cnq 7395   P.cnp 7406   <P cltp 7410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-iota 5233  df-fv 5280  df-iltp 7585

This theorem is referenced by:  nqprl  7666  nqpru  7667  ltprordil  7704  ltnqpr  7708  ltnqpri  7709  ltpopr  7710  ltsopr  7711  ltaddpr  7712  ltexprlemm  7715  ltexprlemopu  7718  ltexprlemru  7727  aptiprleml  7754  aptiprlemu  7755  archpr  7758  cauappcvgprlem2  7775  caucvgprlem2  7795  caucvgprprlemopu  7814  caucvgprprlemexbt  7821  caucvgprprlem2  7825  suplocexprlemloc  7836  suplocexprlemub  7838  suplocexprlemlub  7839