Theorem releldm 5813

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5670 and brv 5356. (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 5602	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5603	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 488	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5778	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1373	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2110  Vcvv 3408   class class class wbr 5053  dom cdm 5551  Rel wrel 5556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-dm 5561

This theorem is referenced by:  releldmb  5815  releldmi  5817  sofld  6050  funeu  6405  fnbr  6486  funbrfv2b  6770  funfvbrb  6871  ercl  8402  inviso1  17271  setciso  17597  lmle  24198  dvidlem  24812  dvmulbr  24836  dvcobr  24843  ulmcau  25287  ulmdvlem3  25294  metideq  31557  heibor1lem  35704  rrncmslem  35727  eqvrelcl  36462  ntrclsiex  41340  ntrneiiex  41363  binomcxplemnn0  41640  binomcxplemnotnn0  41647  sumnnodd  42846  climlimsup  42976  climlimsupcex  42985  climliminflimsupd  43017  liminflimsupclim  43023  dmclimxlim  43067  xlimclimdm  43070  xlimresdm  43075  ioodvbdlimc1lem2  43148  ioodvbdlimc2lem  43150  funbrafv  44322  funbrafv2b  44323  rngciso  45213  rngcisoALTV  45225  ringciso  45264  ringcisoALTV  45288