Theorem releldm 5893

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5749 and brv 5420. (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 5677	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5678	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 484	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5858	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1373	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  Vcvv 3440   class class class wbr 5098  dom cdm 5624  Rel wrel 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634

This theorem is referenced by:  releldmb  5895  releldmi  5897  sofld  6145  funeu  6517  fnbr  6600  funbrfv2b  6891  funfvbrb  6996  ercl  8646  inviso1  17690  setciso  18015  rngciso  20571  ringciso  20605  lmle  25257  dvidlem  25872  dvmulbr  25897  dvmulbrOLD  25898  dvcobr  25905  dvcobrOLD  25906  ulmcau  26360  ulmdvlem3  26367  metideq  34050  heibor1lem  38010  rrncmslem  38033  eqvrelcl  38869  ntrclsiex  44294  ntrneiiex  44317  binomcxplemnn0  44590  binomcxplemnotnn0  44597  sumnnodd  45876  climlimsup  46004  climlimsupcex  46013  climliminflimsupd  46045  liminflimsupclim  46051  dmclimxlim  46095  xlimclimdm  46098  xlimresdm  46103  ioodvbdlimc1lem2  46176  ioodvbdlimc2lem  46178  funbrafv  47404  funbrafv2b  47405  rngcisoALTV  48523  ringcisoALTV  48557  isinito3  49745