Theorem dmcoss2 38439

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 38438	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5657	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2756	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1540  ^◡ccnv 5645  dom cdm 5646  ran crn 5647   ≀ ccoss 38166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-br 5116  df-opab 5178  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-coss 38396

This theorem is referenced by:  dm1cosscnvepres  38441