Theorem f1oi 5674

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):𝐴–1-1-onto→𝐴

Proof of Theorem f1oi

Step	Hyp	Ref	Expression
1		fnresi 5496	. 2 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		cnvresid 5450	. . . 4 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
3	2	fneq1i 5470	. . 3 ⊢ (◡( I ↾ 𝐴) Fn 𝐴 ↔ ( I ↾ 𝐴) Fn 𝐴)
4	1, 3	mpbir 146	. 2 ⊢ ◡( I ↾ 𝐴) Fn 𝐴
5		dff1o4 5642	. 2 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ◡( I ↾ 𝐴) Fn 𝐴))
6	1, 4, 5	mpbir2an 955	1 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴

Syntax hints:   I cid 4428  ^◡ccnv 4768   ↾ cres 4771   Fn wfn 5367  –1-1-onto→wf1o 5371

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379

This theorem is referenced by:  f1ovi  5675  isoid  6006  enrefg  7040  ssdomg  7055  omp1eomlem  7424  ctm  7439  omct  7447  ctssexmid  7480  ssomct  13314  idmhm  13753  idghm  14039  gzsumgsum1  14130  ssidcn  15234  dvid  15719  dvidre  15721  dvexp  15735  ausgrusgrben  16323  subctctexmid  16944