Theorem dmcoss2 37835

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 37834	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5680	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2757	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1533  ^◡ccnv 5668  dom cdm 5669  ran crn 5670   ≀ ccoss 37554

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-coss 37792

This theorem is referenced by:  dm1cosscnvepres  37837