Theorem brrelex2i 4763

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 4760	. 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 2200   _Vcvv 2799   class class class wbr 4083   Rel wrel 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726

This theorem is referenced by:  vtoclr  4767  brdomi  6898  xpdom2  6990  xpdom1g  6992  mapdom1g  7008  djudom  7260  difinfsn  7267  enomnilem  7305  enmkvlem  7328  enwomnilem  7336  djuenun  7394  aprcl  8793  hashinfom  11000  clim  11792  ntrivcvgap0  12060