Theorem dmcoss2 39583

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 39582	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5661	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2786	1 ⊢ dom ≀ 𝑅 = ran 𝑅

This proof depends on syntax axioms:   = wceq 1569  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   ≀ ccoss 39222

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-coss 39540

This theorem is used by:  dm1cosscnvepres  39585