Theorem brrelex12 4731

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 4700	. . . . 5 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 120	. . . 4 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	ssbrd 4102	. . 3 |- ( Rel R -> ( A R B -> A ( _V X. _V ) B ) )
4	3	imp 124	. 2 |- ( ( Rel R /\ A R B ) -> A ( _V X. _V ) B )
5		brxp 4724	. 2 |- ( A ( _V X. _V ) B <-> ( A e. _V /\ B e. _V ) )
6	4, 5	sylib 122	1 |- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2178   _Vcvv 2776   C_ wss 3174   class class class wbr 4059   X. cxp 4691   Rel wrel 4698

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700

This theorem is referenced by:  brrelex1  4732  brrelex  4733  brrelex2  4734  brrelex12i  4735  relbrcnvg  5080  ovprc  6003  ersym  6655  relelec  6685  encv  6856  dvdsrd  13971