Theorem dmcoss2 39300

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 39299	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5670	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2788	1 ⊢ dom ≀ 𝑅 = ran 𝑅

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   ≀ ccoss 38939

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 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-coss 39257

This theorem is used by:  dm1cosscnvepres  39302