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Theorem fcoi2 6777
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 6558 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 cores 6260 . . 3 (ran 𝐹𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹))
3 fnrel 6662 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
4 coi2 6274 . . . 4 (Rel 𝐹 → ( I ∘ 𝐹) = 𝐹)
53, 4syl 17 . . 3 (𝐹 Fn 𝐴 → ( I ∘ 𝐹) = 𝐹)
62, 5sylan9eqr 2788 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
71, 6sylbi 216 1 (𝐹:𝐴𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wss 3947   I cid 5579  ran crn 5683  cres 5684  ccom 5686  Rel wrel 5687   Fn wfn 6549  wf 6550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-fun 6556  df-fn 6557  df-f 6558
This theorem is referenced by:  fcof1oinvd  7307  mapen  9179  mapfien  9451  hashfacen  14471  hashfacenOLD  14472  cofulid  17909  setccatid  18106  estrccatid  18155  efmndid  18878  efmndmnd  18879  symggrp  19398  f1omvdco2  19446  symggen  19468  psgnunilem1  19491  gsumval3  19905  gsumzf1o  19910  frgpcyg  21571  f1linds  21823  qtophmeo  23812  motgrp  28470  hoico2  31690  fcoinver  32524  fcobij  32636  symgfcoeu  32960  symgcom  32961  pmtrcnel2  32968  cycpmconjs  33034  subfacp1lem5  35012  ltrncoidN  39827  trlcoat  40422  trlcone  40427  cdlemg47a  40433  cdlemg47  40435  trljco  40439  tgrpgrplem  40448  tendo1mul  40469  tendo0pl  40490  cdlemkid2  40623  cdlemk45  40646  cdlemk53b  40655  erng1r  40694  tendocnv  40720  dvalveclem  40724  dva0g  40726  dvhgrp  40806  dvhlveclem  40807  dvh0g  40810  cdlemn8  40903  dihordlem7b  40914  dihopelvalcpre  40947  aks6d1c6lem5  41875  mendring  42853  rngccatidALTV  47649  ringccatidALTV  47683
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