Theorem brrelex2i 4817

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 4814	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3	1, 2	mpan 428	1 |- ( A R B -> B e. _V )

Syntax hints:   -> wi 4   e. wcel 2209   _Vcvv 2821   class class class wbr 4128   Rel wrel 4777

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779

This theorem is referenced by:  vtoclr  4821  brdomi  7026  xpdom2  7122  xpdom1g  7124  mapdom1g  7140  suppeqfsuppbi  7288  djudom  7426  difinfsn  7433  enomnilem  7471  enmkvlem  7494  enwomnilem  7502  djuenun  7561  aprcl  8967  hashinfom  11198  clim  12028  ntrivcvgap0  12297