Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  refrelcosslem Structured version   Visualization version   GIF version

Theorem refrelcosslem 36970
Description: Lemma for the left side of the refrelcoss3 36971 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.)
Assertion
Ref Expression
refrelcosslem 𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥

Proof of Theorem refrelcosslem
StepHypRef Expression
1 ralel 3064 . 2 𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅
2 eldmcoss2 36967 . . . 4 (𝑥 ∈ V → (𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥))
32elv 3450 . . 3 (𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥)
43ralbii 3093 . 2 (∀𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅 ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥)
51, 4mpbi 229 1 𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2107  wral 3061  Vcvv 3444   class class class wbr 5106  dom cdm 5634  ccoss 36680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-coss 36919
This theorem is referenced by:  refrelcoss3  36971  eqvrelcoss3  37126
  Copyright terms: Public domain W3C validator