Theorem f1oi 2825

Description: A restriction of the identity relation is a one-to-one onto function.

Assertion

Ref	Expression
f1oi	|- (I |` A):A-1-1-onto->A

Proof of Theorem f1oi

Step	Hyp	Ref	Expression
1		fnresi 2737	. . . . 5 |- (I |` A) Fn A
2		rnresi 2614	. . . . 5 |- ran (I |` A) = A
3	1, 2	pm3.2i 234	. . . 4 |- ((I |` A) Fn A /\ ran (I |` A) = A)
4		df-fo 2436	. . . 4 |- ((I |` A):A-onto->A <-> ((I |` A) Fn A /\ ran (I |` A) = A))
5	3, 4	mpbir 165	. . 3 |- (I |` A):A-onto->A
6		funi 2692	. . . . 5 |- Fun I
7		cnvi 2634	. . . . . 6 |- `'I = I
8		funeq 2683	. . . . . 6 |- (`'I = I -> (Fun `'I <-> Fun I))
9	7, 8	ax-mp 6	. . . . 5 |- (Fun `'I <-> Fun I)
10	6, 9	mpbir 165	. . . 4 |- Fun `'I
11		funres11 2709	. . . 4 |- (Fun `'I -> Fun `'(I |` A))
12	10, 11	ax-mp 6	. . 3 |- Fun `'(I |` A)
13	5, 12	pm3.2i 234	. 2 |- ((I |` A):A-onto->A /\ Fun `'(I |` A))
14		f1o3 2805	. 2 |- ((I |` A):A-1-1-onto->A <-> ((I |` A):A-onto->A /\ Fun `'(I |` A)))
15	13, 14	mpbir 165	1 |- (I |` A):A-1-1-onto->A

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091  Icid 2057  `'ccnv 2409  ran crn 2411   |` cres 2412  Fun wfun 2416   Fn wfn 2417  -onto->wfo 2420  -1-1-onto->wf1o 2421

This theorem is referenced by:  f1ovi 2826  isoid 2933  enrefg 3294  idssen 3309  ssdomg 3311

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-rex 1206  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-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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