Theorem dmcoss2 36551

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 36550	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5599	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2770	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1541  ^◡ccnv 5587  dom cdm 5588  ran crn 5589   ≀ ccoss 36312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-coss 36516

This theorem is referenced by:  dm1cosscnvepres  36553