Theorem brrelex1i 4820

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 4816	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
3	1, 2	mpan 428	1 |- ( A R B -> A e. _V )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   _Vcvv 2821   class class class wbr 4131   Rel wrel 4781

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-xp 4782  df-rel 4783

This theorem is used by:  nprrel  4822  vtoclr  4825  opeliunxp2  4922  ideqg  4933  issetid  4936  fvmptss2  5783  opeliunxp2f  6514  brtpos2  6527  brdomg  7038  ctex  7043  isfi  7053  domssr  7070  en1uniel  7097  xpdom2  7135  xpdom1g  7137  xpen  7151  isbth  7291  relprcnfsupp  7295  djudom  7439  cc3  7640  aprcl  8982  climcl  12072  climi  12077  climrecl  12114  structex  13421