MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fcoi2 Structured version   Unicode version

Theorem fcoi2 5742
Description: Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fcoi2  |-  ( F : A --> B  -> 
( (  _I  |`  B )  o.  F )  =  F )

Proof of Theorem fcoi2
StepHypRef Expression
1 df-f 5574 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 cores 5493 . . 3  |-  ( ran 
F  C_  B  ->  ( (  _I  |`  B )  o.  F )  =  (  _I  o.  F
) )
3 fnrel 5661 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
4 coi2 5507 . . . 4  |-  ( Rel 
F  ->  (  _I  o.  F )  =  F )
53, 4syl 16 . . 3  |-  ( F  Fn  A  ->  (  _I  o.  F )  =  F )
62, 5sylan9eqr 2517 . 2  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  ->  ( (  _I  |`  B )  o.  F )  =  F )
71, 6sylbi 195 1  |-  ( F : A --> B  -> 
( (  _I  |`  B )  o.  F )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    C_ wss 3461    _I cid 4779   ran crn 4989    |` cres 4990    o. ccom 4992   Rel wrel 4993    Fn wfn 5565   -->wf 5566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-fun 5572  df-fn 5573  df-f 5574
This theorem is referenced by:  fcof1oinvd  6171  mapen  7674  mapfien  7859  mapfienOLD  8129  hashfacen  12487  cofulid  15378  setccatid  15562  estrccatid  15600  symggrp  16624  f1omvdco2  16672  symggen  16694  psgnunilem1  16717  gsumval3OLD  17107  gsumval3  17110  gsumzf1o  17116  gsumzf1oOLD  17119  frgpcyg  18785  f1linds  19027  qtophmeo  20484  motgrp  24131  hoico2  26874  fcoinver  27674  fcobij  27779  subfacp1lem5  28892  mendring  31382  rngccatidALTV  33051  ringccatidALTV  33114  ltrncoidN  36249  trlcoat  36846  trlcone  36851  cdlemg47a  36857  cdlemg47  36859  trljco  36863  tgrpgrplem  36872  tendo1mul  36893  tendo0pl  36914  cdlemkid2  37047  cdlemk45  37070  cdlemk53b  37079  erng1r  37118  tendocnv  37145  dvalveclem  37149  dva0g  37151  dvhgrp  37231  dvhlveclem  37232  dvh0g  37235  cdlemn8  37328  dihordlem7b  37339  dihopelvalcpre  37372
  Copyright terms: Public domain W3C validator