Theorem brrelex12 4399

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

Assertion

Ref	Expression
brrelex12	|- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )

Proof of Theorem brrelex12

Step	Hyp	Ref	Expression
1		df-rel 4370	. . . . 5 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 118	. . . 4 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	ssbrd 3826	. . 3 |- ( Rel R -> ( A R B -> A ( _V X. _V ) B ) )
4	3	imp 122	. 2 |- ( ( Rel R /\ A R B ) -> A ( _V X. _V ) B )
5		brxp 4393	. 2 |- ( A ( _V X. _V ) B <-> ( A e. _V /\ B e. _V ) )
6	4, 5	sylib 120	1 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   _Vcvv 2601   C_ wss 2973   class class class wbr 3785   X. cxp 4361   Rel wrel 4368

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370

This theorem is referenced by:  brrelex  4400  brrelex2  4401  relbrcnvg  4724  ovprc  5560  ersym  6141  relelec  6169  encv  6250