Theorem brrelex1i 4706

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 4702	. 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 4033   Rel wrel 4668

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 4151  ax-pow 4207  ax-pr 4242

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 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670

This theorem is referenced by:  nprrel  4708  vtoclr  4711  opeliunxp2  4806  ideqg  4817  issetid  4820  fvmptss2  5636  opeliunxp2f  6296  brtpos2  6309  brdomg  6807  ctex  6812  isfi  6820  en1uniel  6863  xpdom2  6890  xpdom1g  6892  xpen  6906  isbth  7033  djudom  7159  cc3  7335  aprcl  8673  climcl  11447  climi  11452  climrecl  11489  structex  12690