# Metamath Proof Explorer

## Theorem rbropapd

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

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

### Proof

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