Theorem brrelex2i 4685

Description: The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Hypothesis

Ref	Expression
brrelexi.1	|- Rel R

Assertion

Ref	Expression
brrelex2i	|- ( A R B -> B e. _V )

Proof of Theorem brrelex2i

Step	Hyp	Ref	Expression
1		brrelexi.1	. 2 |- Rel R
2		brrelex2 4682	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3	1, 2	mpan 424	1 |- ( A R B -> B e. _V )

Syntax hints:   -> wi 4   e. wcel 2160   _Vcvv 2752   class class class wbr 4018   Rel wrel 4646

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 4136  ax-pow 4189  ax-pr 4224

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 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648

This theorem is referenced by:  vtoclr  4689  brdomi  6770  xpdom2  6852  xpdom1g  6854  mapdom1g  6870  djudom  7117  difinfsn  7124  enomnilem  7161  enmkvlem  7184  enwomnilem  7192  djuenun  7236  aprcl  8628  hashinfom  10785  clim  11316  ntrivcvgap0  11584