Theorem releldm 5908

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5763 and brv 5432. (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 5691	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5692	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 484	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5873	. 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 395   ∈ wcel 2109  Vcvv 3447   class class class wbr 5107  dom cdm 5638  Rel wrel 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-dm 5648

This theorem is referenced by:  releldmb  5910  releldmi  5912  sofld  6160  funeu  6541  fnbr  6626  funbrfv2b  6918  funfvbrb  7023  ercl  8682  inviso1  17728  setciso  18053  rngciso  20547  ringciso  20581  lmle  25201  dvidlem  25816  dvmulbr  25841  dvmulbrOLD  25842  dvcobr  25849  dvcobrOLD  25850  ulmcau  26304  ulmdvlem3  26311  metideq  33883  heibor1lem  37803  rrncmslem  37826  eqvrelcl  38603  ntrclsiex  44042  ntrneiiex  44065  binomcxplemnn0  44338  binomcxplemnotnn0  44345  sumnnodd  45628  climlimsup  45758  climlimsupcex  45767  climliminflimsupd  45799  liminflimsupclim  45805  dmclimxlim  45849  xlimclimdm  45852  xlimresdm  45857  ioodvbdlimc1lem2  45930  ioodvbdlimc2lem  45932  funbrafv  47159  funbrafv2b  47160  rngcisoALTV  48265  ringcisoALTV  48299  isinito3  49489