Theorem brrelex2i 4796

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 4793	. 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 2205   _Vcvv 2815   class class class wbr 4111   Rel wrel 4756

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758

This theorem is referenced by:  vtoclr  4800  brdomi  6988  xpdom2  7084  xpdom1g  7086  mapdom1g  7102  suppeqfsuppbi  7250  djudom  7386  difinfsn  7393  enomnilem  7431  enmkvlem  7454  enwomnilem  7462  djuenun  7521  aprcl  8922  hashinfom  11145  clim  11970  ntrivcvgap0  12239