Theorem isprmpt2 5858

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 3765	. . . 4 |- ( F M P <-> <. F , P >. e. M )
2		isprmpt2.1	. . . . . 6 |- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )
3	2	adantr 261	. . . . 5 |- ( ( ph /\ ( F e. X /\ P e. Y ) ) -> M = { <. f , p >. | ( f W p /\ ps ) } )
4	3	eleq2d 2107	. . . 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 181	. . 3 |- ( ( ph /\ ( F e. X /\ P e. Y ) ) -> ( F M P <-> <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } ) )
6		breq12 3769	. . . . . 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 442	. . . . 5 |- ( ( f = F /\ p = P ) -> ( ( f W p /\ ps ) <-> ( F W P /\ ch ) ) )
9	8	opelopabga 4000	. . . 4 |- ( ( F e. X /\ P e. Y ) -> ( <. F , P >. e. { <. f , p >. | ( f W p /\ ps ) } <-> ( F W P /\ ch ) ) )
10	9	adantl 262	. . 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 177	. 2 |- ( ( ph /\ ( F e. X /\ P e. Y ) ) -> ( F M P <-> ( F W P /\ ch ) ) )
12	11	ex 108	1 |- ( ph -> ( ( F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   <.cop 3378   class class class wbr 3764   {copab 3817

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819

This theorem is referenced by: (None)