Theorem releldm 5969

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5824 and brv 5492. (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 5753	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5754	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 484	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5934	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1371	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  dom cdm 5700  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dm 5710

This theorem is referenced by:  releldmb  5971  releldmi  5973  sofld  6218  funeu  6603  fnbr  6687  funbrfv2b  6979  funfvbrb  7084  ercl  8774  inviso1  17827  setciso  18158  rngciso  20660  ringciso  20694  lmle  25354  dvidlem  25970  dvmulbr  25995  dvmulbrOLD  25996  dvcobr  26003  dvcobrOLD  26004  ulmcau  26456  ulmdvlem3  26463  metideq  33839  heibor1lem  37769  rrncmslem  37792  eqvrelcl  38568  ntrclsiex  44015  ntrneiiex  44038  binomcxplemnn0  44318  binomcxplemnotnn0  44325  sumnnodd  45551  climlimsup  45681  climlimsupcex  45690  climliminflimsupd  45722  liminflimsupclim  45728  dmclimxlim  45772  xlimclimdm  45775  xlimresdm  45780  ioodvbdlimc1lem2  45853  ioodvbdlimc2lem  45855  funbrafv  47073  funbrafv2b  47074  rngcisoALTV  48000  ringcisoALTV  48034