Theorem rbropapd 6245

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 4006	. . . 4 |- ( F M P <-> <. F , P >. e. M )
2		rbropapd.1	. . . . 5 |- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )
3	2	eleq2d 2247	. . . 4 |- ( ph -> ( <. F , P >. e. M <-> <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } ) )
4	1, 3	bitrid 192	. . 3 |- ( ph -> ( F M P <-> <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } ) )
5		breq12 4010	. . . . 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 473	. . . 4 |- ( ( f = F /\ p = P ) -> ( ( f W p /\ ps ) <-> ( F W P /\ ch ) ) )
8	7	opelopabga 4265	. . 3 |- ( ( F e. X /\ P e. Y ) -> ( <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } <-> ( F W P /\ ch ) ) )
9	4, 8	sylan9bb 462	. 2 |- ( ( ph /\ ( F e. X /\ P e. Y ) ) -> ( F M P <-> ( F W P /\ ch ) ) )
10	9	ex 115	1 |- ( ph -> ( ( F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   <.cop 3597   class class class wbr 4005   {copab 4065

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067

This theorem is referenced by:  rbropap  6246