Theorem imacnvcnv 5462

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 5460	. . 3 |- ( `' `' A |` B ) = ( A |` B )
2	1	rneqi 5219	. 2 |- ran ( `' `' A |` B ) = ran ( A |` B )
3		df-ima 5002	. 2 |- ( `' `' A " B ) = ran ( `' `' A |` B )
4		df-ima 5002	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2482	1 |- ( `' `' A " B ) = ( A " B )

Syntax hints:   = wceq 1383   `'ccnv 4988   ran crn 4990   |` cres 4991   "cima 4992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002

This theorem is referenced by:  curry1  6877  curry2  6880  fnwelem  6900  mapfienOLD  8141  fpwwe2lem6  9016  fpwwe2lem9  9019  eqglact  16231  hmeoima  20244  hmeocld  20246  hmeocls  20247  hmeontr  20248  reghmph  20272  qtopf1  20295  tgpconcompeqg  20588  imasf1obl  20969  mbfimaopnlem  22040  hmeoclda  30127