Theorem resiima 5023

Description: The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)

Assertion

Ref	Expression
resiima	|- ( B C_ A -> ( ( _I |` A ) " B ) = B )

Proof of Theorem resiima

Step	Hyp	Ref	Expression
1		df-ima 4672	. . 3 |- ( ( _I |` A ) " B ) = ran ( ( _I |` A ) |` B )
2	1	a1i 9	. 2 |- ( B C_ A -> ( ( _I |` A ) " B ) = ran ( ( _I |` A ) |` B ) )
3		resabs1 4971	. . 3 |- ( B C_ A -> ( ( _I |` A ) |` B ) = ( _I |` B ) )
4	3	rneqd 4891	. 2 |- ( B C_ A -> ran ( ( _I |` A ) |` B ) = ran ( _I |` B ) )
5		rnresi 5022	. . 3 |- ran ( _I |` B ) = B
6	5	a1i 9	. 2 |- ( B C_ A -> ran ( _I |` B ) = B )
7	2, 4, 6	3eqtrd 2230	1 |- ( B C_ A -> ( ( _I |` A ) " B ) = B )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3153   _I cid 4319   ran crn 4660   |` cres 4661   "cima 4662

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 4147  ax-pow 4203  ax-pr 4238

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 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672

This theorem is referenced by:  ssidcn  14378