Theorem brrelex1i 4775

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 4771	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
3	1, 2	mpan 424	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2202  Vcvv 2803   class class class wbr 4093  Rel wrel 4736

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738

This theorem is referenced by:  nprrel  4777  vtoclr  4780  opeliunxp2  4876  ideqg  4887  issetid  4890  fvmptss2  5730  opeliunxp2f  6447  brtpos2  6460  brdomg  6962  ctex  6967  isfi  6977  domssr  6994  en1uniel  7021  xpdom2  7058  xpdom1g  7060  xpen  7074  isbth  7209  djudom  7335  cc3  7530  aprcl  8868  climcl  11905  climi  11910  climrecl  11947  structex  13157