Theorem dmcoss2 35698

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 35697	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5569	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2850	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1536  ^◡ccnv 5557  dom cdm 5558  ran crn 5559   ≀ ccoss 35457

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-coss 35663

This theorem is referenced by:  dm1cosscnvepres  35700