Theorem dmcnvcnv 5911

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 6176 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 5873	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5658	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2784	1 ⊢ dom ◡◡𝐴 = dom 𝐴

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5646  dom cdm 5647  ran crn 5648

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 2732  ax-sep 5248  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-cnv 5655  df-dm 5657  df-rn 5658

This theorem is used by:  resdm2  6221  f1cnvcnv  6777  trrelsuperrel2dg  44615