Theorem brrelex2 4688

Description: A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
brrelex2	|- ( ( Rel R /\ A R B ) -> B e. _V )

Proof of Theorem brrelex2

Step	Hyp	Ref	Expression
1		brrelex12 4685	. 2 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )
2	1	simprd 114	1 |- ( ( Rel R /\ A R B ) -> B e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2160   _Vcvv 2752   class class class wbr 4021   Rel wrel 4652

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-xp 4653  df-rel 4654

This theorem is referenced by:  brrelex2i  4691  releldm  4883  relelrn  4884  elrelimasn  5015  funbrfv  5578  ertr  6578  erth  6609