Theorem resiima 4282

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 4007	. . 3 |- (( _I |` A)"B) = ran (( _I |` A) |` B)
2	1	a1i 8	. 2 |- (B C_ A -> (( _I |` A)"B) = ran (( _I |` A) |` B))
3		resabs1 4244	. . 3 |- (B C_ A -> (( _I |` A) |` B) = ( _I |` B))
4	3	rneqd 4188	. 2 |- (B C_ A -> ran (( _I |` A) |` B) = ran ( _I |` B))
5		rnresi 4281	. . 3 |- ran ( _I |` B) = B
6	5	a1i 8	. 2 |- (B C_ A -> ran ( _I |` B) = B)
7	2, 4, 6	3eqtrd 1929	1 |- (B C_ A -> (( _I |` A)"B) = B)

Syntax hints:   -> wi 3   = wceq 1298   C_ wss 2593   _I cid 3582  ran crn 3987   |` cres 3988  "cima 3989

This theorem is referenced by:  idcn 9042  idhme 14879  hmphre 14884  filfm 15600

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007

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