Theorem dmcoss2 39171

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 39170	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5674	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2789	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1570  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   ≀ ccoss 38810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-coss 39128

This theorem is referenced by:  dm1cosscnvepres  39173