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Theorem f1oi 5107
 Description: A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
f1oi ( I ↾ A):A1-1-ontoA

Proof of Theorem f1oi
StepHypRef Expression
1 fnresi 4959 . 2 ( I ↾ A) Fn A
2 cnvresid 4916 . . . 4 ( I ↾ A) = ( I ↾ A)
32fneq1i 4936 . . 3 (( I ↾ A) Fn A ↔ ( I ↾ A) Fn A)
41, 3mpbir 134 . 2 ( I ↾ A) Fn A
5 dff1o4 5077 . 2 (( I ↾ A):A1-1-ontoA ↔ (( I ↾ A) Fn A ( I ↾ A) Fn A))
61, 4, 5mpbir2an 848 1 ( I ↾ A):A1-1-ontoA
 Colors of variables: wff set class Syntax hints:   I cid 4016  ◡ccnv 4287   ↾ cres 4290   Fn wfn 4840  –1-1-onto→wf1o 4844 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852 This theorem is referenced by:  f1ovi  5108  isoid  5393  enrefg  6180  ssdomg  6194
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