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Theorem refrelcosslem 35233
 Description: Lemma for the left side of the refrelcoss3 35234 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.)
Assertion
Ref Expression
refrelcosslem 𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥

Proof of Theorem refrelcosslem
StepHypRef Expression
1 ralel 3116 . 2 𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅
2 eldmcoss2 35230 . . . 4 (𝑥 ∈ V → (𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥))
32elv 3442 . . 3 (𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥)
43ralbii 3132 . 2 (∀𝑥 ∈ dom ≀ 𝑅𝑥 ∈ dom ≀ 𝑅 ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥)
51, 4mpbi 231 1 𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∈ wcel 2081  ∀wral 3105  Vcvv 3437   class class class wbr 4962  dom cdm 5443   ≀ ccoss 34985 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-coss 35190 This theorem is referenced by:  refrelcoss3  35234  eqvrelcoss3  35384
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