Theorem imadmres 5121

Description: The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
imadmres	|- ( A " dom ( A |` B ) ) = ( A " B )

Proof of Theorem imadmres

Step	Hyp	Ref	Expression
1		resdmres 5120	. . 3 |- ( A |` dom ( A |` B ) ) = ( A |` B )
2	1	rneqi 4855	. 2 |- ran ( A |` dom ( A |` B ) ) = ran ( A |` B )
3		df-ima 4639	. 2 |- ( A " dom ( A |` B ) ) = ran ( A |` dom ( A |` B ) )
4		df-ima 4639	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2208	1 |- ( A " dom ( A |` B ) ) = ( A " B )

Syntax hints:   = wceq 1353   dom cdm 4626   ran crn 4627   |` cres 4628   "cima 4629

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639

This theorem is referenced by:  ssimaex  5577