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Theorem releldm 5778
 Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5637 and brv 5329. (Contributed by NM, 2-Jul-2008.)
Assertion
Ref Expression
releldm ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Proof of Theorem releldm
StepHypRef Expression
1 brrelex1 5569 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
2 brrelex2 5570 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
3 simpr 488 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4 breldmg 5742 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
51, 2, 3, 4syl3anc 1368 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  Vcvv 3441   class class class wbr 5030  dom cdm 5519  Rel wrel 5524 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-dm 5529 This theorem is referenced by:  releldmb  5780  releldmi  5782  sofld  6011  funeu  6349  fnbr  6430  funbrfv2b  6698  funfvbrb  6798  ercl  8285  inviso1  17030  setciso  17345  lmle  23912  dvidlem  24525  dvmulbr  24549  dvcobr  24556  ulmcau  24997  ulmdvlem3  25004  metideq  31258  heibor1lem  35263  rrncmslem  35286  eqvrelcl  36023  ntrclsiex  40771  ntrneiiex  40794  binomcxplemnn0  41068  binomcxplemnotnn0  41075  sumnnodd  42287  climlimsup  42417  climlimsupcex  42426  climliminflimsupd  42458  liminflimsupclim  42464  dmclimxlim  42508  xlimclimdm  42511  xlimresdm  42516  ioodvbdlimc1lem2  42589  ioodvbdlimc2lem  42591  funbrafv  43729  funbrafv2b  43730  rngciso  44621  rngcisoALTV  44633  ringciso  44672  ringcisoALTV  44696
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