Theorem dmcoss2 38152

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 38151	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5693	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2757	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1534  ^◡ccnv 5681  dom cdm 5682  ran crn 5683   ≀ ccoss 37876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-coss 38109

This theorem is referenced by:  dm1cosscnvepres  38154