Theorem releldm 5920

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5772 and brv 5440. (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 5700	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5701	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 488	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5885	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1390	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142  Vcvv 3454   class class class wbr 5100  dom cdm 5647  Rel wrel 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-dm 5657

This theorem is referenced by:  releldmb  5922  releldmi  5924  sofld  6173  funeu  6546  fnbr  6629  funbrfv2b  6924  funfvbrb  7032  ercl  8690  inviso1  17799  setciso  18124  rngciso  20688  ringciso  20722  lmle  25363  dvidlem  25977  dvmulbr  26001  dvcobr  26008  ulmcau  26458  ulmdvlem3  26465  metideq  34190  heibor1lem  38308  rrncmslem  38331  eqvrelcl  39195  ntrclsiex  44629  ntrneiiex  44652  binomcxplemnn0  44925  binomcxplemnotnn0  44932  sumnnodd  46206  climlimsup  46334  climlimsupcex  46343  climliminflimsupd  46375  liminflimsupclim  46381  dmclimxlim  46425  xlimclimdm  46428  xlimresdm  46433  ioodvbdlimc1lem2  46506  ioodvbdlimc2lem  46508  funbrafv  47752  funbrafv2b  47753  rngcisoALTV  48899  ringcisoALTV  48933  isinito3  50121