Theorem brrelex1i 4762

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	⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)

Proof of Theorem brrelex1i

Step	Hyp	Ref	Expression
1		brrelexi.1	. 2 ⊢ Rel 𝑅
2		brrelex1 4758	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
3	1, 2	mpan 424	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2200  Vcvv 2799   class class class wbr 4083  Rel wrel 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726

This theorem is referenced by:  nprrel  4764  vtoclr  4767  opeliunxp2  4862  ideqg  4873  issetid  4876  fvmptss2  5711  opeliunxp2f  6390  brtpos2  6403  brdomg  6905  ctex  6910  isfi  6920  domssr  6937  en1uniel  6964  xpdom2  6998  xpdom1g  7000  xpen  7014  isbth  7145  djudom  7271  cc3  7465  aprcl  8804  climcl  11808  climi  11813  climrecl  11850  structex  13059