Theorem releldm 5279

Description: The first argument of a binary relation belongs to its domain. (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		brrelex 5080	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5081	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 476	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5252	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1318	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  dom cdm 5038  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048

This theorem is referenced by:  releldmb  5281  releldmi  5283  sofld  5500  funeu  5828  fnbr  5907  funbrfv2b  6150  funfvbrb  6238  ercl  7640  inviso1  16249  setciso  16564  lmle  22907  dvidlem  23485  dvmulbr  23508  dvcobr  23515  ulmcau  23953  ulmdvlem3  23960  uhgraun  25840  umgraun  25857  metideq  29264  heibor1lem  32778  rrncmslem  32801  ntrclsiex  37371  ntrneiiex  37394  binomcxplemnn0  37570  binomcxplemnotnn0  37577  sumnnodd  38697  ioodvbdlimc1lem2  38822  ioodvbdlimc2lem  38824  funbrafv  39887  funbrafv2b  39888  rngciso  41774  rngcisoALTV  41786  ringciso  41825  ringcisoALTV  41849