Theorem resiima 3419

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

Assertion

Ref	Expression
resiima	|- (B (_ A -> ((I |` A)"B) = B)

Proof of Theorem resiima

Step	Hyp	Ref	Expression
1		df-ima 3191	. . 3 |- ((I |` A)"B) = ran ((I |` A) |` B)
2	1	a1i 8	. 2 |- (B (_ A -> ((I |` A)"B) = ran ((I |` A) |` B))
3		resabs1 3388	. . 3 |- (B (_ A -> ((I |` A) |` B) = (I |` B))
4	3	rneqd 3341	. 2 |- (B (_ A -> ran ((I |` A) |` B) = ran ( I |` B))
5		rnresi 3418	. . 3 |- ran ( I |` B) = B
6	5	a1i 8	. 2 |- (B (_ A -> ran ( I |` B) = B)
7	2, 4, 6	3eqtrd 1511	1 |- (B (_ A -> ((I |` A)"B) = B)

Syntax hints:   -> wi 3   = wceq 956   (_ wss 2047  Icid 2831  ran crn 3171   |` cres 3172  "cima 3173

This theorem is referenced by:  idcn 7766  idhme 10522  hmphre 10530

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

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