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Theorem releldm 4864
Description: The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
Assertion
Ref Expression
releldm ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Proof of Theorem releldm
StepHypRef Expression
1 brrelex 4668 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
2 brrelex2 4669 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
3 simpr 110 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4 breldmg 4835 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
51, 2, 3, 4syl3anc 1238 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2148  Vcvv 2739   class class class wbr 4005  dom cdm 4628  Rel wrel 4633
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-dm 4638
This theorem is referenced by:  releldmb  4866  releldmi  4868  funeu  5243  fnbr  5320  relelfvdm  5549  funbrfv2b  5563  funfvbrb  5632  ercl  6549  dvidlemap  14300  dvmulxxbr  14306  dviaddf  14309  dvimulf  14310  dvcoapbr  14311
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