Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  imacnvcnv Structured version   Visualization version   GIF version

Theorem imacnvcnv 6062
 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 6060 . . 3 (𝐴𝐵) = (𝐴𝐵)
21rneqi 5806 . 2 ran (𝐴𝐵) = ran (𝐴𝐵)
3 df-ima 5567 . 2 (𝐴𝐵) = ran (𝐴𝐵)
4 df-ima 5567 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2859 1 (𝐴𝐵) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1530  ◡ccnv 5553  ran crn 5555   ↾ cres 5556   “ cima 5557 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5560  df-rel 5561  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567 This theorem is referenced by:  curry1  7795  curry2  7798  fnwelem  7821  fpwwe2lem6  10051  fpwwe2lem9  10054  eqglact  18276  hmeoima  22308  hmeocld  22310  hmeocls  22311  hmeontr  22312  reghmph  22336  qtopf1  22359  tgpconncompeqg  22654  imasf1obl  23032  mbfimaopnlem  24190  hmeoclda  33584
 Copyright terms: Public domain W3C validator