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Theorem fcoi2 5977
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:𝐴𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)

Proof of Theorem fcoi2
StepHypRef Expression
1 df-f 5794 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 cores 5541 . . 3 (ran 𝐹𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹))
3 fnrel 5889 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
4 coi2 5555 . . . 4 (Rel 𝐹 → ( I ∘ 𝐹) = 𝐹)
53, 4syl 17 . . 3 (𝐹 Fn 𝐴 → ( I ∘ 𝐹) = 𝐹)
62, 5sylan9eqr 2666 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
71, 6sylbi 206 1 (𝐹:𝐴𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wss 3540   I cid 4938  ran crn 5029  cres 5030  ccom 5032  Rel wrel 5033   Fn wfn 5785  wf 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4579  df-opab 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-fun 5792  df-fn 5793  df-f 5794
This theorem is referenced by:  fcof1oinvd  6426  mapen  7987  mapfien  8174  hashfacen  13050  cofulid  16322  setccatid  16506  estrccatid  16544  symggrp  17592  f1omvdco2  17640  symggen  17662  psgnunilem1  17685  gsumval3  18080  gsumzf1o  18085  frgpcyg  19689  f1linds  19931  qtophmeo  21378  motgrp  25184  hoico2  27794  fcoinver  28592  fcobij  28682  symgfcoeu  28970  subfacp1lem5  30214  ltrncoidN  34226  trlcoat  34823  trlcone  34828  cdlemg47a  34834  cdlemg47  34836  trljco  34840  tgrpgrplem  34849  tendo1mul  34870  tendo0pl  34891  cdlemkid2  35024  cdlemk45  35047  cdlemk53b  35056  erng1r  35095  tendocnv  35122  dvalveclem  35126  dva0g  35128  dvhgrp  35208  dvhlveclem  35209  dvh0g  35212  cdlemn8  35305  dihordlem7b  35316  dihopelvalcpre  35349  mendring  36575  rngccatidALTV  41773  ringccatidALTV  41836
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