Theorem resiima 5040

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 4688	. . 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 4988	. . 3 |- ( B C_ A -> ( ( _I |` A ) |` B ) = ( _I |` B ) )
4	3	rneqd 4907	. 2 |- ( B C_ A -> ran ( ( _I |` A ) |` B ) = ran ( _I |` B ) )
5		rnresi 5039	. . 3 |- ran ( _I |` B ) = B
6	5	a1i 9	. 2 |- ( B C_ A -> ran ( _I |` B ) = B )
7	2, 4, 6	3eqtrd 2242	1 |- ( B C_ A -> ( ( _I |` A ) " B ) = B )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3166   _I cid 4335   ran crn 4676   |` cres 4677   "cima 4678

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688

This theorem is referenced by:  ssidcn  14682