Theorem brrelex2 4478

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 4475	. 2 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )
2	1	simprd 112	1 |- ( ( Rel R /\ A R B ) -> B e. _V )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1438   _Vcvv 2619   class class class wbr 3845   Rel wrel 4443

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445

This theorem is referenced by:  brrelex2i  4482  releldm  4670  relelrn  4671  elreimasng  4798  funbrfv  5343  ertr  6307  erth  6336