Theorem dmcoss2 37930

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 37929	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5691	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2758	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1533  ^◡ccnv 5679  dom cdm 5680  ran crn 5681   ≀ ccoss 37653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-coss 37887

This theorem is referenced by:  dm1cosscnvepres  37932