Theorem imadmres 5159

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 5158	. . 3 |- ( A |` dom ( A |` B ) ) = ( A |` B )
2	1	rneqi 4891	. 2 |- ran ( A |` dom ( A |` B ) ) = ran ( A |` B )
3		df-ima 4673	. 2 |- ( A " dom ( A |` B ) ) = ran ( A |` dom ( A |` B ) )
4		df-ima 4673	. 2 |- ( A " B ) = ran ( A |` B )
5	2, 3, 4	3eqtr4i 2224	1 |- ( A " dom ( A |` B ) ) = ( A " B )

Syntax hints:   = wceq 1364   dom cdm 4660   ran crn 4661   |` cres 4662   "cima 4663

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673

This theorem is referenced by:  ssimaex  5619