Theorem imacnvcnv 5470

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 5468	. . 3 |- ( `' `' A |` B ) = ( A |` B )
2	1	rneqi 5227	. 2 |- ran ( `' `' A |` B ) = ran ( A |` B )
3		df-ima 5012	. 2 |- ( `' `' A " B ) = ran ( `' `' A |` B )
4		df-ima 5012	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2506	1 |- ( `' `' A " B ) = ( A " B )

Syntax hints:   = wceq 1379   `'ccnv 4998   ran crn 5000   |` cres 5001   "cima 5002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012

This theorem is referenced by:  curry1  6872  curry2  6875  fnwelem  6895  mapfienOLD  8134  fpwwe2lem6  9009  fpwwe2lem9  9012  eqglact  16044  hmeoima  19998  hmeocld  20000  hmeocls  20001  hmeontr  20002  reghmph  20026  qtopf1  20049  tgpconcompeqg  20342  imasf1obl  20723  mbfimaopnlem  21794  hmeoclda  29726