Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dmcoss2 Structured version   Visualization version   GIF version

Theorem dmcoss2 35853
 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 35852 . 2 dom ≀ 𝑅 = dom 𝑅
2 df-rn 5534 . 2 ran 𝑅 = dom 𝑅
31, 2eqtr4i 2827 1 dom ≀ 𝑅 = ran 𝑅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ◡ccnv 5522  dom cdm 5523  ran crn 5524   ≀ ccoss 35612 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-coss 35818 This theorem is referenced by:  dm1cosscnvepres  35855
 Copyright terms: Public domain W3C validator