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Theorem dmcoss3 37957
Description: The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.)
Assertion
Ref Expression
dmcoss3 dom ≀ 𝑅 = dom 𝑅

Proof of Theorem dmcoss3
StepHypRef Expression
1 dfcoss3 37918 . . 3 𝑅 = (𝑅𝑅)
21dmeqi 5911 . 2 dom ≀ 𝑅 = dom (𝑅𝑅)
3 rncnv 37804 . . . 4 ran 𝑅 = dom 𝑅
43eqimssi 4042 . . 3 ran 𝑅 ⊆ dom 𝑅
5 dmcosseq 5980 . . 3 (ran 𝑅 ⊆ dom 𝑅 → dom (𝑅𝑅) = dom 𝑅)
64, 5ax-mp 5 . 2 dom (𝑅𝑅) = dom 𝑅
72, 6eqtri 2756 1 dom ≀ 𝑅 = dom 𝑅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wss 3949  ccnv 5681  dom cdm 5682  ran crn 5683  ccom 5686  ccoss 37681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-coss 37915
This theorem is referenced by:  dmcoss2  37958  eldmcoss  37962
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