Theorem rbropap 6452

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 6451	. 2 |- ( ph -> ( ( F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) ) )
4	3	3impib 1228	1 |- ( ( ph /\ F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093   {copab 4154

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156

This theorem is referenced by: (None)