ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1oi GIF version

Theorem f1oi 5511
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 5345 . 2 ( I ↾ 𝐴) Fn 𝐴
2 cnvresid 5302 . . . 4 ( I ↾ 𝐴) = ( I ↾ 𝐴)
32fneq1i 5322 . . 3 (( I ↾ 𝐴) Fn 𝐴 ↔ ( I ↾ 𝐴) Fn 𝐴)
41, 3mpbir 146 . 2 ( I ↾ 𝐴) Fn 𝐴
5 dff1o4 5481 . 2 (( I ↾ 𝐴):𝐴1-1-onto𝐴 ↔ (( I ↾ 𝐴) Fn 𝐴( I ↾ 𝐴) Fn 𝐴))
61, 4, 5mpbir2an 943 1 ( I ↾ 𝐴):𝐴1-1-onto𝐴
Colors of variables: wff set class
Syntax hints:   I cid 4300  ccnv 4637  cres 4640   Fn wfn 5223  1-1-ontowf1o 5227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235
This theorem is referenced by:  f1ovi  5512  isoid  5824  enrefg  6778  ssdomg  6792  omp1eomlem  7107  ctm  7122  omct  7130  ctssexmid  7162  ssomct  12460  idmhm  12882  idghm  13153  ssidcn  14063  dvid  14515  dvexp  14528  subctctexmid  15104
  Copyright terms: Public domain W3C validator