Theorem dmcnvcnv 5922

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 5884	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5671	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2786	1 ⊢ dom ◡◡𝐴 = dom 𝐴

This proof depends on syntax axioms:   = wceq 1569  ^◡ccnv 5659  dom cdm 5660  ran crn 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-cnv 5668  df-dm 5670  df-rn 5671

This theorem is used by:  resdm2  6231  f1cnvcnv  6785  trrelsuperrel2dg  44425