Theorem dmcnvcnv 5924

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 6177 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 5887	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5680	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2760	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1541  ^◡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 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-cnv 5677  df-dm 5679  df-rn 5680

This theorem is referenced by:  resdm2  6219  f1cnvcnv  6784  trrelsuperrel2dg  42193