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Theorem dmcoss2 36962
Description: The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.)
Assertion
Ref Expression
dmcoss2 dom ≀ 𝑅 = ran 𝑅

Proof of Theorem dmcoss2
StepHypRef Expression
1 dmcoss3 36961 . 2 dom ≀ 𝑅 = dom 𝑅
2 df-rn 5645 . 2 ran 𝑅 = dom 𝑅
31, 2eqtr4i 2764 1 dom ≀ 𝑅 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  ccnv 5633  dom cdm 5634  ran crn 5635  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:  dm1cosscnvepres  36964
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