Theorem brrelexi 4476

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
brrelexi	|- ( A R B -> A e. _V )

Proof of Theorem brrelexi

Step	Hyp	Ref	Expression
1		brrelexi.1	. 2 |- Rel R
2		brrelex 4474	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
3	1, 2	mpan 415	1 |- ( A R B -> A e. _V )

Syntax hints:   -> wi 4   e. wcel 1438   _Vcvv 2619   class class class wbr 3843   Rel wrel 4441

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-br 3844  df-opab 3898  df-xp 4442  df-rel 4443

This theorem is referenced by:  nprrel  4480  vtoclr  4482  opeliunxp2  4572  ideqg  4583  issetid  4586  fvmptss2  5373  brtpos2  6008  brdomg  6455  isfi  6468  en1uniel  6511  xpdom2  6537  xpdom1g  6539  xpen  6551  isbth  6666  djudom  6777  climcl  10657  climi  10662  climrecl  10699