Theorem imacnvcnv 5303

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 5301	. . 3 |- ( `' `' A |` B ) = ( A |` B )
2	1	rneqi 5066	. 2 |- ran ( `' `' A |` B ) = ran ( A |` B )
3		df-ima 4853	. 2 |- ( `' `' A " B ) = ran ( `' `' A |` B )
4		df-ima 4853	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2473	1 |- ( `' `' A " B ) = ( A " B )

Syntax hints:   = wceq 1369   `'ccnv 4839   ran crn 4841   |` cres 4842   "cima 4843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-rel 4847  df-cnv 4848  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853

This theorem is referenced by:  curry1  6664  curry2  6667  fnwelem  6687  mapfienOLD  7927  fpwwe2lem6  8802  fpwwe2lem9  8805  eqglact  15732  hmeoima  19338  hmeocld  19340  hmeocls  19341  hmeontr  19342  reghmph  19366  qtopf1  19389  tgpconcompeqg  19682  imasf1obl  20063  mbfimaopnlem  21133  hmeoclda  28528