Theorem dmcoss2 36258

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 36257	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5547	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2762	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1543  ^◡ccnv 5535  dom cdm 5536  ran crn 5537   ≀ ccoss 36019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-coss 36223

This theorem is referenced by:  dm1cosscnvepres  36260