Theorem dmcoss2 38865

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 38864	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5642	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2762	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1542  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   ≀ ccoss 38504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-coss 38822

This theorem is referenced by:  dm1cosscnvepres  38867