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Theorem imacnvcnv 6057
Description: The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
Assertion
Ref Expression
imacnvcnv (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem imacnvcnv
StepHypRef Expression
1 rescnvcnv 6055 . . 3 (𝐴𝐵) = (𝐴𝐵)
21rneqi 5801 . 2 ran (𝐴𝐵) = ran (𝐴𝐵)
3 df-ima 5562 . 2 (𝐴𝐵) = ran (𝐴𝐵)
4 df-ima 5562 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2854 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  ccnv 5548  ran crn 5550  cres 5551  cima 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562
This theorem is referenced by:  curry1  7790  curry2  7793  fnwelem  7816  fpwwe2lem6  10046  fpwwe2lem9  10049  eqglact  18271  hmeoima  22303  hmeocld  22305  hmeocls  22306  hmeontr  22307  reghmph  22331  qtopf1  22354  tgpconncompeqg  22649  imasf1obl  23027  mbfimaopnlem  24185  hmeoclda  33579
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