Theorem releldm 5935

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5787 and brv 5455. (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 5715	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5716	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 489	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5900	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1396	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149  Vcvv 3463   class class class wbr 5113  dom cdm 5662  Rel wrel 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-dm 5672

This theorem is referenced by:  releldmb  5937  releldmi  5939  sofld  6186  funeu  6562  fnbr  6644  funbrfv2b  6939  funfvbrb  7047  ercl  8705  inviso1  17822  setciso  18147  rngciso  20722  ringciso  20756  lmle  25428  dvidlem  26042  dvmulbr  26066  dvcobr  26073  ulmcau  26523  ulmdvlem3  26530  metideq  34227  heibor1lem  38347  rrncmslem  38370  eqvrelcl  39234  ntrclsiex  44670  ntrneiiex  44693  binomcxplemnn0  44950  binomcxplemnotnn0  44957  sumnnodd  46237  climlimsup  46365  climlimsupcex  46374  climliminflimsupd  46406  liminflimsupclim  46412  dmclimxlim  46456  xlimclimdm  46459  xlimresdm  46464  ioodvbdlimc1lem2  46537  ioodvbdlimc2lem  46539  funbrafv  47783  funbrafv2b  47784  rngcisoALTV  48930  ringcisoALTV  48964  isinito3  50162