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Theorem releldm 5496
Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5383 and brv 5068. (Contributed by NM, 2-Jul-2008.)
Assertion
Ref Expression
releldm ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Proof of Theorem releldm
StepHypRef Expression
1 brrelex 5296 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
2 brrelex2 5297 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
3 simpr 471 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4 breldmg 5468 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
51, 2, 3, 4syl3anc 1475 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2144  Vcvv 3349   class class class wbr 4784  dom cdm 5249  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-dm 5259
This theorem is referenced by:  releldmb  5498  releldmi  5500  sofld  5722  funeu  6056  fnbr  6133  funbrfv2b  6382  funfvbrb  6473  ercl  7906  inviso1  16632  setciso  16947  lmle  23317  dvidlem  23898  dvmulbr  23921  dvcobr  23928  ulmcau  24368  ulmdvlem3  24375  metideq  30270  heibor1lem  33933  rrncmslem  33956  ntrclsiex  38870  ntrneiiex  38893  binomcxplemnn0  39067  binomcxplemnotnn0  39074  sumnnodd  40374  climlimsup  40504  climlimsupcex  40513  climliminflimsupd  40545  liminflimsupclim  40551  ioodvbdlimc1lem2  40659  ioodvbdlimc2lem  40661  funbrafv  41752  funbrafv2b  41753  rngciso  42500  rngcisoALTV  42512  ringciso  42551  ringcisoALTV  42575
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