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Theorem dmcoss3 38358
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 38319 . . 3 𝑅 = (𝑅𝑅)
21dmeqi 5928 . 2 dom ≀ 𝑅 = dom (𝑅𝑅)
3 rncnv 38205 . . . 4 ran 𝑅 = dom 𝑅
43eqimssi 4063 . . 3 ran 𝑅 ⊆ dom 𝑅
5 dmcosseq 5998 . . 3 (ran 𝑅 ⊆ dom 𝑅 → dom (𝑅𝑅) = dom 𝑅)
64, 5ax-mp 5 . 2 dom (𝑅𝑅) = dom 𝑅
72, 6eqtri 2762 1 dom ≀ 𝑅 = dom 𝑅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wss 3970  ccnv 5698  dom cdm 5699  ran crn 5700  ccom 5703  ccoss 38084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-coss 38316
This theorem is referenced by:  dmcoss2  38359  eldmcoss  38363
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