Theorem resiima 5086

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 4732	. . 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 5034	. . 3 |- ( B C_ A -> ( ( _I |` A ) |` B ) = ( _I |` B ) )
4	3	rneqd 4953	. 2 |- ( B C_ A -> ran ( ( _I |` A ) |` B ) = ran ( _I |` B ) )
5		rnresi 5085	. . 3 |- ran ( _I |` B ) = B
6	5	a1i 9	. 2 |- ( B C_ A -> ran ( _I |` B ) = B )
7	2, 4, 6	3eqtrd 2266	1 |- ( B C_ A -> ( ( _I |` A ) " B ) = B )

Syntax hints:   -> wi 4   = wceq 1395   C_ wss 3197   _I cid 4379   ran crn 4720   |` cres 4721   "cima 4722

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by:  ssidcn  14884