Theorem brrelex1i 4813

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 4809	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
3	1, 2	mpan 428	1 |- ( A R B -> A e. _V )

Syntax hints:   -> wi 4   e. wcel 2209   _Vcvv 2821   class class class wbr 4125   Rel wrel 4774

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 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 4244  ax-pow 4306  ax-pr 4341

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776

This theorem is referenced by:  nprrel  4815  vtoclr  4818  opeliunxp2  4915  ideqg  4926  issetid  4929  fvmptss2  5774  opeliunxp2f  6499  brtpos2  6512  brdomg  7022  ctex  7027  isfi  7037  domssr  7054  en1uniel  7081  xpdom2  7119  xpdom1g  7121  xpen  7135  isbth  7274  relprcnfsupp  7278  djudom  7423  cc3  7624  aprcl  8964  climcl  12026  climi  12031  climrecl  12068  structex  13342