Theorem rbropapd 6189

Description: Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)

Hypotheses

Ref	Expression
rbropapd.1	|- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )
rbropapd.2	|- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
rbropapd	|- ( ph -> ( ( F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) ) )

Distinct variable groups:   f, F, p   P, f, p   f, W, p   ch, f, p

Allowed substitution hints:   ph( f, p)   ps( f, p)   M( f, p)   X( f, p)   Y( f, p)

Proof of Theorem rbropapd

Step	Hyp	Ref	Expression
1		df-br 3966	. . . 4 |- ( F M P <-> <. F , P >. e. M )
2		rbropapd.1	. . . . 5 |- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )
3	2	eleq2d 2227	. . . 4 |- ( ph -> ( <. F , P >. e. M <-> <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } ) )
4	1, 3	syl5bb 191	. . 3 |- ( ph -> ( F M P <-> <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } ) )
5		breq12 3970	. . . . 5 |- ( ( f = F /\ p = P ) -> ( f W p <-> F W P ) )
6		rbropapd.2	. . . . 5 |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )
7	5, 6	anbi12d 465	. . . 4 |- ( ( f = F /\ p = P ) -> ( ( f W p /\ ps ) <-> ( F W P /\ ch ) ) )
8	7	opelopabga 4223	. . 3 |- ( ( F e. X /\ P e. Y ) -> ( <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } <-> ( F W P /\ ch ) ) )
9	4, 8	sylan9bb 458	. 2 |- ( ( ph /\ ( F e. X /\ P e. Y ) ) -> ( F M P <-> ( F W P /\ ch ) ) )
10	9	ex 114	1 |- ( ph -> ( ( F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   <.cop 3563   class class class wbr 3965   {copab 4024

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026

This theorem is referenced by:  rbropap  6190