Theorem rbropapd 6328

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 4045	. . . 4 |- ( F M P <-> <. F , P >. e. M )
2		rbropapd.1	. . . . 5 |- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )
3	2	eleq2d 2275	. . . 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 4049	. . . . 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 4309	. . 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 1373   e. wcel 2176   <.cop 3636   class class class wbr 4044   {copab 4104

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-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106

This theorem is referenced by:  rbropap  6329