MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnresi Structured version   Unicode version

Theorem fnresi 5554
Description: Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
Assertion
Ref Expression
fnresi  |-  (  _I  |`  A )  Fn  A

Proof of Theorem fnresi
StepHypRef Expression
1 funi 5475 . . 3  |-  Fun  _I
2 funres 5484 . . 3  |-  ( Fun 
_I  ->  Fun  (  _I  |`  A ) )
31, 2ax-mp 8 . 2  |-  Fun  (  _I  |`  A )
4 dmresi 5188 . 2  |-  dom  (  _I  |`  A )  =  A
5 df-fn 5449 . 2  |-  ( (  _I  |`  A )  Fn  A  <->  ( Fun  (  _I  |`  A )  /\  dom  (  _I  |`  A )  =  A ) )
63, 4, 5mpbir2an 887 1  |-  (  _I  |`  A )  Fn  A
Colors of variables: wff set class
Syntax hints:    = wceq 1652    _I cid 4485   dom cdm 4870    |` cres 4872   Fun wfun 5440    Fn wfn 5441
This theorem is referenced by:  f1oi  5705  fveqf1o  6021  weniso  6067  iordsmo  6611  fipreima  7404  dfac9  8008  ustuqtop3  18265  fta1blem  20083  qaa  20232  dfiop2  23248  cvmliftlem4  24967  cvmliftlem5  24968  fninfp  26726  fndifnfp  26728  fnnfpeq0  26730  pmtrfinv  27370  dvsid  27516  ltrnid  30869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-fun 5448  df-fn 5449
  Copyright terms: Public domain W3C validator