Theorem brrelex1i 4652

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 4648	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
3	1, 2	mpan 422	1 |- ( A R B -> A e. _V )

Syntax hints:   -> wi 4   e. wcel 2141   _Vcvv 2730   class class class wbr 3987   Rel wrel 4614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616

This theorem is referenced by:  nprrel  4654  vtoclr  4657  opeliunxp2  4749  ideqg  4760  issetid  4763  fvmptss2  5569  opeliunxp2f  6215  brtpos2  6228  brdomg  6724  ctex  6729  isfi  6737  en1uniel  6780  xpdom2  6807  xpdom1g  6809  xpen  6821  isbth  6942  djudom  7068  cc3  7223  aprcl  8558  climcl  11238  climi  11243  climrecl  11280  structex  12421