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

metamath.org