Theorem resiima 4956

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 4611	. . 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 4907	. . 3 |- ( B C_ A -> ( ( _I |` A ) |` B ) = ( _I |` B ) )
4	3	rneqd 4827	. 2 |- ( B C_ A -> ran ( ( _I |` A ) |` B ) = ran ( _I |` B ) )
5		rnresi 4955	. . 3 |- ran ( _I |` B ) = B
6	5	a1i 9	. 2 |- ( B C_ A -> ran ( _I |` B ) = B )
7	2, 4, 6	3eqtrd 2201	1 |- ( B C_ A -> ( ( _I |` A ) " B ) = B )

Syntax hints:   -> wi 4   = wceq 1342   C_ wss 3111   _I cid 4260   ran crn 4599   |` cres 4600   "cima 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611

This theorem is referenced by:  ssidcn  12751