Theorem ltdfpr 7619

Description: More convenient form of df-iltp 7583. (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 4045	. . 3 |- ( A <P B <-> <. A , B >. e. <P )
2		df-iltp 7583	. . . 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 5580	. . . . . 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 5580	. . . . . 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 4311	. 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 4044   {copab 4104   ` cfv 5271   1stc1st 6224   2ndc2nd 6225   Q.cnq 7393   P.cnp 7404   <P cltp 7408

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 4162  ax-pow 4218  ax-pr 4253

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 4045  df-opab 4106  df-iota 5232  df-fv 5279  df-iltp 7583

This theorem is referenced by:  nqprl  7664  nqpru  7665  ltprordil  7702  ltnqpr  7706  ltnqpri  7707  ltpopr  7708  ltsopr  7709  ltaddpr  7710  ltexprlemm  7713  ltexprlemopu  7716  ltexprlemru  7725  aptiprleml  7752  aptiprlemu  7753  archpr  7756  cauappcvgprlem2  7773  caucvgprlem2  7793  caucvgprprlemopu  7812  caucvgprprlemexbt  7819  caucvgprprlem2  7823  suplocexprlemloc  7834  suplocexprlemub  7836  suplocexprlemlub  7837