Theorem releldm 5938

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5786 and brv 5448. (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 5711	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5712	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 490	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5903	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1399	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  Vcvv 3454   class class class wbr 5108  dom cdm 5658  Rel wrel 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-dm 5668

This theorem is used by:  releldmb  5940  releldmi  5942  sofld  6197  funeu  6593  fnbr  6677  funbrfv2b  6973  funfvbrb  7081  ercl  8767  inviso1  17991  setciso  18316  rngciso  20941  ringciso  20975  lmle  25677  dvidlem  26290  dvmulbr  26314  dvcobr  26321  ulmcau  26779  ulmdvlem3  26786  metideq  34636  heibor1lem  39009  rrncmslem  39032  eqvrelcl  39894  ntrclsiex  45314  ntrneiiex  45337  binomcxplemnn0  45594  binomcxplemnotnn0  45601  sumnnodd  46915  climlimsup  47043  climlimsupcex  47052  climliminflimsupd  47084  liminflimsupclim  47090  dmclimxlim  47134  xlimclimdm  47137  xlimresdm  47142  ioodvbdlimc1lem2  47215  ioodvbdlimc2lem  47217  funbrafv  48501  funbrafv2b  48502  rngcisoALTV  49647  ringcisoALTV  49681  isinito3  50966