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Theorem fcoi2 6767
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 6548 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 cores 6249 . . 3 (ran 𝐹𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹))
3 fnrel 6652 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
4 coi2 6263 . . . 4 (Rel 𝐹 → ( I ∘ 𝐹) = 𝐹)
53, 4syl 17 . . 3 (𝐹 Fn 𝐴 → ( I ∘ 𝐹) = 𝐹)
62, 5sylan9eqr 2795 . 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 397   = wceq 1542  wss 3949   I cid 5574  ran crn 5678  cres 5679  ccom 5681  Rel wrel 5682   Fn wfn 6539  wf 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  fcof1oinvd  7291  mapen  9141  mapfien  9403  hashfacen  14413  hashfacenOLD  14414  cofulid  17840  setccatid  18034  estrccatid  18083  efmndid  18769  efmndmnd  18770  symggrp  19268  f1omvdco2  19316  symggen  19338  psgnunilem1  19361  gsumval3  19775  gsumzf1o  19780  frgpcyg  21129  f1linds  21380  qtophmeo  23321  motgrp  27794  hoico2  31010  fcoinver  31835  fcobij  31947  symgfcoeu  32243  symgcom  32244  pmtrcnel2  32251  cycpmconjs  32315  subfacp1lem5  34175  ltrncoidN  38999  trlcoat  39594  trlcone  39599  cdlemg47a  39605  cdlemg47  39607  trljco  39611  tgrpgrplem  39620  tendo1mul  39641  tendo0pl  39662  cdlemkid2  39795  cdlemk45  39818  cdlemk53b  39827  erng1r  39866  tendocnv  39892  dvalveclem  39896  dva0g  39898  dvhgrp  39978  dvhlveclem  39979  dvh0g  39982  cdlemn8  40075  dihordlem7b  40086  dihopelvalcpre  40119  mendring  41934  rngccatidALTV  46887  ringccatidALTV  46950
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