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Theorem fcoi2 6783
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 6565 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 cores 6269 . . 3 (ran 𝐹𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹))
3 fnrel 6670 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
4 coi2 6283 . . . 4 (Rel 𝐹 → ( I ∘ 𝐹) = 𝐹)
53, 4syl 17 . . 3 (𝐹 Fn 𝐴 → ( I ∘ 𝐹) = 𝐹)
62, 5sylan9eqr 2799 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
71, 6sylbi 217 1 (𝐹:𝐴𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wss 3951   I cid 5577  ran crn 5686  cres 5687  ccom 5689  Rel wrel 5690   Fn wfn 6556  wf 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  fcof1oinvd  7313  mapen  9181  mapfien  9448  hashfacen  14493  cofulid  17935  setccatid  18129  estrccatid  18176  efmndid  18901  efmndmnd  18902  symggrp  19418  f1omvdco2  19466  symggen  19488  psgnunilem1  19511  gsumval3  19925  gsumzf1o  19930  frgpcyg  21592  f1linds  21845  qtophmeo  23825  motgrp  28551  hoico2  31776  fcoinver  32617  fcobij  32733  symgfcoeu  33102  symgcom  33103  pmtrcnel2  33110  cycpmconjs  33176  subfacp1lem5  35189  ltrncoidN  40130  trlcoat  40725  trlcone  40730  cdlemg47a  40736  cdlemg47  40738  trljco  40742  tgrpgrplem  40751  tendo1mul  40772  tendo0pl  40793  cdlemkid2  40926  cdlemk45  40949  cdlemk53b  40958  erng1r  40997  tendocnv  41023  dvalveclem  41027  dva0g  41029  dvhgrp  41109  dvhlveclem  41110  dvh0g  41113  cdlemn8  41206  dihordlem7b  41217  dihopelvalcpre  41250  aks6d1c6lem5  42178  mendring  43200  rngccatidALTV  48188  ringccatidALTV  48222
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