Theorem dmcoss2 38430

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 38429	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5634	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2755	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1540  ^◡ccnv 5622  dom cdm 5623  ran crn 5624   ≀ ccoss 38154

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-coss 38387

This theorem is referenced by:  dm1cosscnvepres  38432