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Theorem dmcoss3 36961
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 36922 . . 3 𝑅 = (𝑅𝑅)
21dmeqi 5861 . 2 dom ≀ 𝑅 = dom (𝑅𝑅)
3 rncnv 36807 . . . 4 ran 𝑅 = dom 𝑅
43eqimssi 4003 . . 3 ran 𝑅 ⊆ dom 𝑅
5 dmcosseq 5929 . . 3 (ran 𝑅 ⊆ dom 𝑅 → dom (𝑅𝑅) = dom 𝑅)
64, 5ax-mp 5 . 2 dom (𝑅𝑅) = dom 𝑅
72, 6eqtri 2761 1 dom ≀ 𝑅 = dom 𝑅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wss 3911  ccnv 5633  dom cdm 5634  ran crn 5635  ccom 5638  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-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:  dmcoss2  36962  eldmcoss  36966
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