Theorem isprmpt2 5881

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

Hypotheses

Ref	Expression
isprmpt2.1	|- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )
isprmpt2.2	|- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
isprmpt2	|- ( 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 isprmpt2

Step	Hyp	Ref	Expression
1		df-br 3786	. . . 4 |- ( F M P <-> <. F , P >. e. M )
2		isprmpt2.1	. . . . . 6 |- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )
3	2	adantr 270	. . . . 5 |- ( ( ph /\ ( F e. X /\ P e. Y ) ) -> M = { <. f , p >. | ( f W p /\ ps ) } )
4	3	eleq2d 2148	. . . 4 |- ( ( ph /\ ( F e. X /\ P e. Y ) ) -> ( <. F , P >. e. M <-> <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } ) )
5	1, 4	syl5bb 190	. . 3 |- ( ( ph /\ ( F e. X /\ P e. Y ) ) -> ( F M P <-> <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } ) )
6		breq12 3790	. . . . . 6 |- ( ( f = F /\ p = P ) -> ( f W p <-> F W P ) )
7		isprmpt2.2	. . . . . 6 |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )
8	6, 7	anbi12d 456	. . . . 5 |- ( ( f = F /\ p = P ) -> ( ( f W p /\ ps ) <-> ( F W P /\ ch ) ) )
9	8	opelopabga 4018	. . . 4 |- ( ( F e. X /\ P e. Y ) -> ( <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } <-> ( F W P /\ ch ) ) )
10	9	adantl 271	. . 3 |- ( ( ph /\ ( F e. X /\ P e. Y ) ) -> ( <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } <-> ( F W P /\ ch ) ) )
11	5, 10	bitrd 186	. 2 |- ( ( ph /\ ( F e. X /\ P e. Y ) ) -> ( F M P <-> ( F W P /\ ch ) ) )
12	11	ex 113	1 |- ( ph -> ( ( F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   <.cop 3401   class class class wbr 3785   {copab 3838

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840

This theorem is referenced by: (None)