Theorem imacnvcnv 5319

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

Assertion

Ref	Expression
imacnvcnv	|- ( `' `' A " B ) = ( A " B )

Proof of Theorem imacnvcnv

Step	Hyp	Ref	Expression
1		rescnvcnv 5317	. . 3 |- ( `' `' A |` B ) = ( A |` B )
2	1	rneqi 5080	. 2 |- ran ( `' `' A |` B ) = ran ( A |` B )
3		df-ima 4866	. 2 |- ( `' `' A " B ) = ran ( `' `' A |` B )
4		df-ima 4866	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2494	1 |- ( `' `' A " B ) = ( A " B )

Syntax hints:   = wceq 1455   `'ccnv 4852   ran crn 4854   |` cres 4855   "cima 4856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-xp 4859  df-rel 4860  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866

This theorem is referenced by:  curry1  6915  curry2  6918  fnwelem  6938  fpwwe2lem6  9086  fpwwe2lem9  9089  eqglact  16917  hmeoima  20829  hmeocld  20831  hmeocls  20832  hmeontr  20833  reghmph  20857  qtopf1  20880  tgpconcompeqg  21175  imasf1obl  21552  mbfimaopnlem  22660  hmeoclda  31038