Theorem releldm 5842

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5699 and brv 5381. (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 5631	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5632	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 484	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5807	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1369	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070  dom cdm 5580  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-dm 5590

This theorem is referenced by:  releldmb  5844  releldmi  5846  sofld  6079  funeu  6443  fnbr  6525  funbrfv2b  6809  funfvbrb  6910  ercl  8467  inviso1  17395  setciso  17722  lmle  24370  dvidlem  24984  dvmulbr  25008  dvcobr  25015  ulmcau  25459  ulmdvlem3  25466  metideq  31745  heibor1lem  35894  rrncmslem  35917  eqvrelcl  36652  ntrclsiex  41552  ntrneiiex  41575  binomcxplemnn0  41856  binomcxplemnotnn0  41863  sumnnodd  43061  climlimsup  43191  climlimsupcex  43200  climliminflimsupd  43232  liminflimsupclim  43238  dmclimxlim  43282  xlimclimdm  43285  xlimresdm  43290  ioodvbdlimc1lem2  43363  ioodvbdlimc2lem  43365  funbrafv  44537  funbrafv2b  44538  rngciso  45428  rngcisoALTV  45440  ringciso  45479  ringcisoALTV  45503