Theorem imacnvcnv 5229

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 5227	. . 3 |- ( `' `' A |` B ) = ( A |` B )
2	1	rneqi 4987	. 2 |- ran ( `' `' A |` B ) = ran ( A |` B )
3		df-ima 4764	. 2 |- ( `' `' A " B ) = ran ( `' `' A |` B )
4		df-ima 4764	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2265	1 |- ( `' `' A " B ) = ( A " B )

Syntax hints:   = wceq 1398   `'ccnv 4750   ran crn 4752   |` cres 4753   "cima 4754

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-cnv 4759  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764

This theorem is referenced by:  casedm  7379  caseinl  7384  caseinr  7385  djudm  7398  eqglact  13959  hmeoima  15192  hmeocld  15194  hmeontr  15195