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Theorem refrelcosslem 38443
Description: Lemma for the left side of the refrelcoss3 38444 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.)
Assertion
Ref Expression
refrelcosslem 𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥

Proof of Theorem refrelcosslem
StepHypRef Expression
1 ralel 3061 . 2 𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅
2 eldmcoss2 38440 . . . 4 (𝑥 ∈ V → (𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥))
32elv 3482 . . 3 (𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥)
43ralbii 3090 . 2 (∀𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅 ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥)
51, 4mpbi 230 1 𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2105  wral 3058  Vcvv 3477   class class class wbr 5147  dom cdm 5688  ccoss 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-coss 38392
This theorem is referenced by:  refrelcoss3  38444  eqvrelcoss3  38599
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