Theorem ltdfpr 7573

Description: More convenient form of df-iltp 7537. (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 4034	. . 3 |- ( A <P B <-> <. A , B >. e. <P )
2		df-iltp 7537	. . . 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 2263	. . 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 5562	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( 2nd ` x ) = ( 2nd ` A ) )
7	6	eleq2d 2266	. . . . 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 5562	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( 1st ` y ) = ( 1st ` B ) )
10	9	eleq2d 2266	. . . . 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 2498	. . 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 4299	. 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 1364   e. wcel 2167   E.wrex 2476   <.cop 3625   class class class wbr 4033   {copab 4093   ` cfv 5258   1stc1st 6196   2ndc2nd 6197   Q.cnq 7347   P.cnp 7358   <P cltp 7362

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-iota 5219  df-fv 5266  df-iltp 7537

This theorem is referenced by:  nqprl  7618  nqpru  7619  ltprordil  7656  ltnqpr  7660  ltnqpri  7661  ltpopr  7662  ltsopr  7663  ltaddpr  7664  ltexprlemm  7667  ltexprlemopu  7670  ltexprlemru  7679  aptiprleml  7706  aptiprlemu  7707  archpr  7710  cauappcvgprlem2  7727  caucvgprlem2  7747  caucvgprprlemopu  7766  caucvgprprlemexbt  7773  caucvgprprlem2  7777  suplocexprlemloc  7788  suplocexprlemub  7790  suplocexprlemlub  7791