Theorem fnresi 3609

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 3199	. 2 |- ((I |` A) Fn A <-> (Fun (I |` A) /\ dom ( I |` A) = A))
2		funi 3551	. . 3 |- Fun I
3		funres 3557	. . 3 |- (Fun I -> Fun (I |` A))
4	2, 3	ax-mp 7	. 2 |- Fun (I |` A)
5		dmresi 3405	. 2 |- dom ( I |` A) = A
6	1, 4, 5	mpbir2an 732	1 |- (I |` A) Fn A

Syntax hints:   = wceq 958  Icid 2837  dom cdm 3176   |` cres 3178  Fun wfun 3182   Fn wfn 3183

This theorem is referenced by:  f1oi 3723  idcn 7763  dfiop2 9674

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-res 3196  df-fun 3198  df-fn 3199

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