Theorem releldm 5901

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5757 and brv 5428. (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 5685	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5686	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 484	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5866	. 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 3442   class class class wbr 5100  dom cdm 5632  Rel wrel 5637

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 2709  ax-sep 5243  ax-pr 5379

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-dm 5642

This theorem is referenced by:  releldmb  5903  releldmi  5905  sofld  6153  funeu  6525  fnbr  6608  funbrfv2b  6899  funfvbrb  7005  ercl  8657  inviso1  17702  setciso  18027  rngciso  20583  ringciso  20617  lmle  25269  dvidlem  25884  dvmulbr  25909  dvmulbrOLD  25910  dvcobr  25917  dvcobrOLD  25918  ulmcau  26372  ulmdvlem3  26379  metideq  34071  heibor1lem  38060  rrncmslem  38083  eqvrelcl  38947  ntrclsiex  44409  ntrneiiex  44432  binomcxplemnn0  44705  binomcxplemnotnn0  44712  sumnnodd  45990  climlimsup  46118  climlimsupcex  46127  climliminflimsupd  46159  liminflimsupclim  46165  dmclimxlim  46209  xlimclimdm  46212  xlimresdm  46217  ioodvbdlimc1lem2  46290  ioodvbdlimc2lem  46292  funbrafv  47518  funbrafv2b  47519  rngcisoALTV  48637  ringcisoALTV  48671  isinito3  49859