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Theorem releldm 5266
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 5070 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
2 brrelex2 5071 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
3 simpr 475 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4 breldmg 5239 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
51, 2, 3, 4syl3anc 1317 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1976  Vcvv 3172   class class class wbr 4577  dom cdm 5028  Rel wrel 5033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5034  df-rel 5035  df-dm 5038
This theorem is referenced by:  releldmb  5268  releldmi  5270  sofld  5486  funeu  5814  fnbr  5893  funbrfv2b  6135  funfvbrb  6223  ercl  7617  inviso1  16195  setciso  16510  lmle  22825  dvidlem  23402  dvmulbr  23425  dvcobr  23432  ulmcau  23870  ulmdvlem3  23877  uhgraun  25606  umgraun  25623  metideq  29070  heibor1lem  32574  rrncmslem  32597  ntrclsiex  37167  ntrneiiex  37190  binomcxplemnn0  37366  binomcxplemnotnn0  37373  sumnnodd  38494  ioodvbdlimc1lem2  38619  ioodvbdlimc2lem  38621  funbrafv  39685  funbrafv2b  39686  rngciso  41769  rngcisoALTV  41781  ringciso  41820  ringcisoALTV  41844
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