Theorem dmcoss2 38450

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 38449	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5704	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2768	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1539  ^◡ccnv 5692  dom cdm 5693  ran crn 5694   ≀ ccoss 38176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5152  df-opab 5214  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-coss 38407

This theorem is referenced by:  dm1cosscnvepres  38452