Theorem dmcoss2 39234

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 39233	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5677	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2792	1 ⊢ dom ≀ 𝑅 = ran 𝑅

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   ≀ ccoss 38873

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-coss 39191

This theorem is used by:  dm1cosscnvepres  39236