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Theorem f1oi 5405
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 5240 . 2 ( I ↾ 𝐴) Fn 𝐴
2 cnvresid 5197 . . . 4 ( I ↾ 𝐴) = ( I ↾ 𝐴)
32fneq1i 5217 . . 3 (( I ↾ 𝐴) Fn 𝐴 ↔ ( I ↾ 𝐴) Fn 𝐴)
41, 3mpbir 145 . 2 ( I ↾ 𝐴) Fn 𝐴
5 dff1o4 5375 . 2 (( I ↾ 𝐴):𝐴1-1-onto𝐴 ↔ (( I ↾ 𝐴) Fn 𝐴( I ↾ 𝐴) Fn 𝐴))
61, 4, 5mpbir2an 926 1 ( I ↾ 𝐴):𝐴1-1-onto𝐴
Colors of variables: wff set class
Syntax hints:   I cid 4210  ccnv 4538  cres 4541   Fn wfn 5118  1-1-ontowf1o 5122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130
This theorem is referenced by:  f1ovi  5406  isoid  5711  enrefg  6658  ssdomg  6672  omp1eomlem  6979  ctm  6994  omct  7002  ctssexmid  7024  ssidcn  12389  dvid  12841  dvexp  12854  subctctexmid  13226
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