Theorem dmcnvcnv 5918

Description: The domain of the double converse of a class is equal to its domain (even when that class in not a relation, in which case dfrel2 6183 gives another proof). (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
dmcnvcnv	⊢ dom ◡◡𝐴 = dom 𝐴

Proof of Theorem dmcnvcnv

Step	Hyp	Ref	Expression
1		dfdm4 5880	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5670	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2760	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1540  ^◡ccnv 5658  dom cdm 5659  ran crn 5660

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is referenced by:  resdm2  6225  f1cnvcnv  6788  trrelsuperrel2dg  43662