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Theorem releldm 4496
Description: The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
Assertion
Ref Expression
releldm ((Rel 𝑅 A𝑅B) → A dom 𝑅)

Proof of Theorem releldm
StepHypRef Expression
1 brrelex 4309 . 2 ((Rel 𝑅 A𝑅B) → A V)
2 brrelex2 4310 . 2 ((Rel 𝑅 A𝑅B) → B V)
3 simpr 103 . 2 ((Rel 𝑅 A𝑅B) → A𝑅B)
4 breldmg 4468 . 2 ((A V B V A𝑅B) → A dom 𝑅)
51, 2, 3, 4syl3anc 1121 1 ((Rel 𝑅 A𝑅B) → A dom 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1374  Vcvv 2535   class class class wbr 3738  dom cdm 4272  Rel wrel 4277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-dm 4282
This theorem is referenced by:  releldmb  4498  releldmi  4500  funeu  4852  fnbr  4927  relelfvdm  5130  funbrfv2b  5143  funfvbrb  5205  ercl  6028
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