Theorem dmcnvcnv 5921

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 6186 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 5883	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5670	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2786	1 ⊢ dom ◡◡𝐴 = dom 𝐴

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5658  dom cdm 5659  ran crn 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is used by:  resdm2  6231  f1cnvcnv  6786  trrelsuperrel2dg  44498