Theorem releldm 5944

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5801 and brv 5473. (Contributed by NM, 2-Jul-2008.)

Assertion

Ref	Expression
releldm	⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Proof of Theorem releldm

Step	Hyp	Ref	Expression
1		brrelex1 5730	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5731	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 486	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5910	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1372	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  Vcvv 3475   class class class wbr 5149  dom cdm 5677  Rel wrel 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dm 5687

This theorem is referenced by:  releldmb  5946  releldmi  5948  sofld  6187  funeu  6574  fnbr  6658  funbrfv2b  6950  funfvbrb  7053  ercl  8714  inviso1  17713  setciso  18041  lmle  24818  dvidlem  25432  dvmulbr  25456  dvcobr  25463  ulmcau  25907  ulmdvlem3  25914  metideq  32873  gg-dvmulbr  35175  gg-dvcobr  35176  heibor1lem  36677  rrncmslem  36700  eqvrelcl  37482  ntrclsiex  42804  ntrneiiex  42827  binomcxplemnn0  43108  binomcxplemnotnn0  43115  sumnnodd  44346  climlimsup  44476  climlimsupcex  44485  climliminflimsupd  44517  liminflimsupclim  44523  dmclimxlim  44567  xlimclimdm  44570  xlimresdm  44575  ioodvbdlimc1lem2  44648  ioodvbdlimc2lem  44650  funbrafv  45866  funbrafv2b  45867  rngciso  46880  rngcisoALTV  46892  ringciso  46931  ringcisoALTV  46955