Theorem rncnvcnv 4607

Description: The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
rncnvcnv	⊢ ran ◡◡𝐴 = ran 𝐴

Proof of Theorem rncnvcnv

Step	Hyp	Ref	Expression
1		df-rn 4402	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		dfdm4 4575	. 2 ⊢ dom ◡𝐴 = ran ◡◡𝐴
3	1, 2	eqtr2i 2104	1 ⊢ ran ◡◡𝐴 = ran 𝐴

Syntax hints:   = wceq 1285  ^◡ccnv 4390  dom cdm 4391  ran crn 4392

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-cnv 4399  df-dm 4401  df-rn 4402

This theorem is referenced by:  rnresv  4830