Theorem dmcoss2 38359

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 38358	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5710	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2765	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1537  ^◡ccnv 5698  dom cdm 5699  ran crn 5700   ≀ ccoss 38084

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 2105  ax-9 2113  ax-10 2136  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450

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 2712  df-cleq 2726  df-clel 2813  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-coss 38316

This theorem is referenced by:  dm1cosscnvepres  38361