Theorem imacnvcnv 5107

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 5105	. . 3 |- ( `' `' A |` B ) = ( A |` B )
2	1	rneqi 4869	. 2 |- ran ( `' `' A |` B ) = ran ( A |` B )
3		df-ima 4653	. 2 |- ( `' `' A " B ) = ran ( `' `' A |` B )
4		df-ima 4653	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2219	1 |- ( `' `' A " B ) = ( A " B )

Syntax hints:   = wceq 1363   `'ccnv 4639   ran crn 4641   |` cres 4642   "cima 4643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018  df-opab 4079  df-xp 4646  df-rel 4647  df-cnv 4648  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653

This theorem is referenced by:  casedm  7102  caseinl  7107  caseinr  7108  djudm  7121  eqglact  13129  hmeoima  14193  hmeocld  14195  hmeontr  14196