Theorem imacnvcnv 5380

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 5378	. . 3 |- ( `' `' A |` B ) = ( A |` B )
2	1	rneqi 5142	. 2 |- ran ( `' `' A |` B ) = ran ( A |` B )
3		df-ima 4926	. 2 |- ( `' `' A " B ) = ran ( `' `' A |` B )
4		df-ima 4926	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2421	1 |- ( `' `' A " B ) = ( A " B )

Syntax hints:   = wceq 1399   `'ccnv 4912   ran crn 4914   |` cres 4915   "cima 4916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926

This theorem is referenced by:  curry1  6791  curry2  6794  fnwelem  6814  mapfienOLD  8051  fpwwe2lem6  8924  fpwwe2lem9  8927  eqglact  16369  hmeoima  20351  hmeocld  20353  hmeocls  20354  hmeontr  20355  reghmph  20379  qtopf1  20402  tgpconcompeqg  20695  imasf1obl  21076  mbfimaopnlem  22147  hmeoclda  30317