Theorem brrelex2i 4699

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 4696	. 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 2164   _Vcvv 2760   class class class wbr 4029   Rel wrel 4660

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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4661  df-rel 4662

This theorem is referenced by:  vtoclr  4703  brdomi  6794  xpdom2  6876  xpdom1g  6878  mapdom1g  6894  djudom  7142  difinfsn  7149  enomnilem  7187  enmkvlem  7210  enwomnilem  7218  djuenun  7262  aprcl  8655  hashinfom  10839  clim  11414  ntrivcvgap0  11682