Theorem fnresi 2737

Description: Functionality and domain of restricted identity.

Assertion

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

Proof of Theorem fnresi

Step	Hyp	Ref	Expression
1		funi 2692	. . . 4 |- Fun I
2		funres 2697	. . . 4 |- (Fun I -> Fun (I |` A))
3	1, 2	ax-mp 6	. . 3 |- Fun (I |` A)
4		ssv 1520	. . . . 5 |- A (_ V
5		dmi 2545	. . . . 5 |- dom I = V
6	4, 5	sseqtr4 1533	. . . 4 |- A (_ dom I
7		ssdmres 2585	. . . 4 |- (A (_ dom I <-> dom (I |` A) = A)
8	6, 7	mpbi 164	. . 3 |- dom (I |` A) = A
9	3, 8	pm3.2i 234	. 2 |- (Fun (I |` A) /\ dom (I |` A) = A)
10		df-fn 2433	. 2 |- ((I |` A) Fn A <-> (Fun (I |` A) /\ dom (I |` A) = A))
11	9, 10	mpbir 165	1 |- (I |` A) Fn A

Syntax hints:   /\ wa 196   = wceq 1091  Vcvv 1348   (_ wss 1487  Icid 2057  dom cdm 2410   |` cres 2412  Fun wfun 2416   Fn wfn 2417

This theorem is referenced by:  f1oi 2825  ho1 5613

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-res 2430  df-fun 2432  df-fn 2433

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