Theorem brrelex1i 4707

Description: The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)

Hypothesis

Ref	Expression
brrelexi.1	|- Rel R

Assertion

Ref	Expression
brrelex1i	|- ( A R B -> A e. _V )

Proof of Theorem brrelex1i

Step	Hyp	Ref	Expression
1		brrelexi.1	. 2 |- Rel R
2		brrelex1 4703	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
3	1, 2	mpan 424	1 |- ( A R B -> A e. _V )

Syntax hints:   -> wi 4   e. wcel 2167   _Vcvv 2763   class class class wbr 4034   Rel wrel 4669

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671

This theorem is referenced by:  nprrel  4709  vtoclr  4712  opeliunxp2  4807  ideqg  4818  issetid  4821  fvmptss2  5639  opeliunxp2f  6305  brtpos2  6318  brdomg  6816  ctex  6821  isfi  6829  en1uniel  6872  xpdom2  6899  xpdom1g  6901  xpen  6915  isbth  7042  djudom  7168  cc3  7351  aprcl  8690  climcl  11464  climi  11469  climrecl  11506  structex  12715