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Theorem dmcoss3 38411
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 38372 . . 3 𝑅 = (𝑅𝑅)
21dmeqi 5929 . 2 dom ≀ 𝑅 = dom (𝑅𝑅)
3 rncnv 38258 . . . 4 ran 𝑅 = dom 𝑅
43eqimssi 4069 . . 3 ran 𝑅 ⊆ dom 𝑅
5 dmcosseq 6001 . . 3 (ran 𝑅 ⊆ dom 𝑅 → dom (𝑅𝑅) = dom 𝑅)
64, 5ax-mp 5 . 2 dom (𝑅𝑅) = dom 𝑅
72, 6eqtri 2768 1 dom ≀ 𝑅 = dom 𝑅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wss 3976  ccnv 5699  dom cdm 5700  ran crn 5701  ccom 5704  ccoss 38137
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 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
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 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-coss 38369
This theorem is referenced by:  dmcoss2  38412  eldmcoss  38416
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