Theorem releldm 5899

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5756 and brv 5425. (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 5684	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5685	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 484	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5864	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1374	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3429   class class class wbr 5085  dom cdm 5631  Rel wrel 5636

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-dm 5641

This theorem is referenced by:  releldmb  5901  releldmi  5903  sofld  6151  funeu  6523  fnbr  6606  funbrfv2b  6897  funfvbrb  7003  ercl  8655  inviso1  17733  setciso  18058  rngciso  20615  ringciso  20649  lmle  25268  dvidlem  25882  dvmulbr  25906  dvcobr  25913  ulmcau  26360  ulmdvlem3  26367  metideq  34037  heibor1lem  38130  rrncmslem  38153  eqvrelcl  39017  ntrclsiex  44480  ntrneiiex  44503  binomcxplemnn0  44776  binomcxplemnotnn0  44783  sumnnodd  46060  climlimsup  46188  climlimsupcex  46197  climliminflimsupd  46229  liminflimsupclim  46235  dmclimxlim  46279  xlimclimdm  46282  xlimresdm  46287  ioodvbdlimc1lem2  46360  ioodvbdlimc2lem  46362  funbrafv  47606  funbrafv2b  47607  rngcisoALTV  48753  ringcisoALTV  48787  isinito3  49975