Theorem elopabran 4378

Description: Membership in an ordered-pair class abstraction defined by a restricted binary relation. (Contributed by AV, 16-Feb-2021.)

Assertion

Ref	Expression
elopabran	|- ( A e. { <. x , y >. | ( x R y /\ ps ) } -> A e. R )

Distinct variable groups:   x, R   y, R

Allowed substitution hints:   ps( x, y)   A( x, y)

Proof of Theorem elopabran

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 |- ( ( x R y /\ ps ) -> x R y )
2	1	ssopab2i 4372	. . 3 |- { <. x , y >. | ( x R y /\ ps ) } C_ { <. x , y >. | x R y }
3		opabss 4153	. . 3 |- { <. x , y >. | x R y } C_ R
4	2, 3	sstri 3236	. 2 |- { <. x , y >. | ( x R y /\ ps ) } C_ R
5	4	sseli 3223	1 |- ( A e. { <. x , y >. | ( x R y /\ ps ) } -> A e. R )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   class class class wbr 4088   {copab 4149

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213  df-br 4089  df-opab 4151

This theorem is referenced by:  trlsfvalg  16233