Theorem brrelex1i 4719

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 4715	. 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 2176   _Vcvv 2772   class class class wbr 4045   Rel wrel 4681

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-rel 4683

This theorem is referenced by:  nprrel  4721  vtoclr  4724  opeliunxp2  4819  ideqg  4830  issetid  4833  fvmptss2  5656  opeliunxp2f  6326  brtpos2  6339  brdomg  6839  ctex  6844  isfi  6854  domssr  6871  en1uniel  6898  xpdom2  6928  xpdom1g  6930  xpen  6944  isbth  7071  djudom  7197  cc3  7382  aprcl  8721  climcl  11626  climi  11631  climrecl  11668  structex  12877