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Theorem f1oi 5465
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 5300 . 2 ( I ↾ 𝐴) Fn 𝐴
2 cnvresid 5257 . . . 4 ( I ↾ 𝐴) = ( I ↾ 𝐴)
32fneq1i 5277 . . 3 (( I ↾ 𝐴) Fn 𝐴 ↔ ( I ↾ 𝐴) Fn 𝐴)
41, 3mpbir 145 . 2 ( I ↾ 𝐴) Fn 𝐴
5 dff1o4 5435 . 2 (( I ↾ 𝐴):𝐴1-1-onto𝐴 ↔ (( I ↾ 𝐴) Fn 𝐴( I ↾ 𝐴) Fn 𝐴))
61, 4, 5mpbir2an 931 1 ( I ↾ 𝐴):𝐴1-1-onto𝐴
Colors of variables: wff set class
Syntax hints:   I cid 4261  ccnv 4598  cres 4601   Fn wfn 5178  1-1-ontowf1o 5182
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-opab 4039  df-id 4266  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190
This theorem is referenced by:  f1ovi  5466  isoid  5773  enrefg  6722  ssdomg  6736  omp1eomlem  7051  ctm  7066  omct  7074  ctssexmid  7106  ssomct  12341  ssidcn  12777  dvid  13229  dvexp  13242  subctctexmid  13743
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