Theorem dmcnvcnv 5926

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 6182 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 5889	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5680	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2755	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1533  ^◡ccnv 5668  dom cdm 5669  ran crn 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-cnv 5677  df-dm 5679  df-rn 5680

This theorem is referenced by:  resdm2  6224  f1cnvcnv  6791  trrelsuperrel2dg  42995