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 5672 and brv 5363. (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 5604	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5605	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 487	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5777	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1367	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  Vcvv 3494   class class class wbr 5065  dom cdm 5554  Rel wrel 5559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-dm 5564

This theorem is referenced by:  releldmb  5815  releldmi  5817  sofld  6043  funeu  6379  fnbr  6458  funbrfv2b  6722  funfvbrb  6820  ercl  8299  inviso1  17035  setciso  17350  lmle  23903  dvidlem  24512  dvmulbr  24535  dvcobr  24542  ulmcau  24982  ulmdvlem3  24989  metideq  31133  heibor1lem  35086  rrncmslem  35109  eqvrelcl  35846  ntrclsiex  40401  ntrneiiex  40424  binomcxplemnn0  40679  binomcxplemnotnn0  40686  sumnnodd  41909  climlimsup  42039  climlimsupcex  42048  climliminflimsupd  42080  liminflimsupclim  42086  dmclimxlim  42130  xlimclimdm  42133  xlimresdm  42138  ioodvbdlimc1lem2  42215  ioodvbdlimc2lem  42217  funbrafv  43356  funbrafv2b  43357  rngciso  44252  rngcisoALTV  44264  ringciso  44303  ringcisoALTV  44327