Theorem brrelexi 4410

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 4408	. 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 1434   _Vcvv 2602   class class class wbr 3793   Rel wrel 4376

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378

This theorem is referenced by:  nprrel  4412  vtoclr  4414  opeliunxp2  4504  ideqg  4515  issetid  4518  fvmptss2  5279  brtpos2  5900  brdomg  6295  isfi  6308  en1uniel  6351  xpdom2  6375  xpdom1g  6377  xpen  6386  climcl  10259  climi  10264  climrecl  10300