Theorem dmcoss2 39055

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

Step	Hyp	Ref	Expression
1		dmcoss3 39054	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5663	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2791	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1563  ^◡ccnv 5651  dom cdm 5652  ran crn 5653   ≀ ccoss 38694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-coss 39012

This theorem is referenced by:  dm1cosscnvepres  39057