Theorem fnresi 3603

Description: Functionality and domain of restricted identity.

Assertion

Ref	Expression
fnresi	|- (I |` A) Fn A

Proof of Theorem fnresi

Step	Hyp	Ref	Expression
1		df-fn 3193	. 2 |- ((I |` A) Fn A <-> (Fun (I |` A) /\ dom ( I |` A) = A))
2		funi 3545	. . 3 |- Fun I
3		funres 3551	. . 3 |- (Fun I -> Fun (I |` A))
4	2, 3	ax-mp 7	. 2 |- Fun (I |` A)
5		dmresi 3399	. 2 |- dom ( I |` A) = A
6	1, 4, 5	mpbir2an 730	1 |- (I |` A) Fn A

Syntax hints:   = wceq 956  Icid 2831  dom cdm 3170   |` cres 3172  Fun wfun 3176   Fn wfn 3177

This theorem is referenced by:  f1oi 3717  idcn 7766  dfiop2 9679

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-9 965  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-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-co 3187  df-dm 3188  df-res 3190  df-fun 3192  df-fn 3193

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