Theorem brrelex1i 4669

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 4665	. 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 2148   _Vcvv 2737   class class class wbr 4003   Rel wrel 4631

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633

This theorem is referenced by:  nprrel  4671  vtoclr  4674  opeliunxp2  4767  ideqg  4778  issetid  4781  fvmptss2  5591  opeliunxp2f  6238  brtpos2  6251  brdomg  6747  ctex  6752  isfi  6760  en1uniel  6803  xpdom2  6830  xpdom1g  6832  xpen  6844  isbth  6965  djudom  7091  cc3  7266  aprcl  8602  climcl  11289  climi  11294  climrecl  11331  structex  12473