Theorem dmcnvcnv 5886

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 6150 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 5849	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5642	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2753	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1540  ^◡ccnv 5630  dom cdm 5631  ran crn 5632

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 5246  ax-nul 5256  ax-pr 5382

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by:  resdm2  6192  f1cnvcnv  6747  trrelsuperrel2dg  43653