Theorem rbropap 6331

Description: Properties of a pair in a restricted binary relation M expressed as an ordered-pair class abstraction: M is the binary relation W restricted by the condition ps. (Contributed by AV, 31-Jan-2021.)

Hypotheses

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

Assertion

Ref	Expression
rbropap	|- ( ( 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 rbropap

Step	Hyp	Ref	Expression
1		rbropapd.1	. . 3 |- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )
2		rbropapd.2	. . 3 |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )
3	1, 2	rbropapd 6330	. 2 |- ( ph -> ( ( F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) ) )
4	3	3impib 1204	1 |- ( ( ph /\ F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   class class class wbr 4045   {copab 4105

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 4163  ax-pow 4219  ax-pr 4254

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 4046  df-opab 4107

This theorem is referenced by: (None)