Theorem dmcnvcnv 5943

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 6208 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 5905	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5695	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2765	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1539  ^◡ccnv 5683  dom cdm 5684  ran crn 5685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-cnv 5692  df-dm 5694  df-rn 5695

This theorem is referenced by:  resdm2  6250  f1cnvcnv  6812  trrelsuperrel2dg  43689