Theorem brrelex2i 4630

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 4627	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3	1, 2	mpan 421	1 |- ( A R B -> B e. _V )

Syntax hints:   -> wi 4   e. wcel 2128   _Vcvv 2712   class class class wbr 3965   Rel wrel 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593

This theorem is referenced by:  vtoclr  4634  brdomi  6694  xpdom2  6776  xpdom1g  6778  mapdom1g  6792  djudom  7037  difinfsn  7044  enomnilem  7081  enmkvlem  7104  enwomnilem  7112  djuenun  7147  aprcl  8521  hashinfom  10652  clim  11178  ntrivcvgap0  11446