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Theorem fcoi1 6768
Description: Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fcoi1 (𝐹:𝐴𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)

Proof of Theorem fcoi1
StepHypRef Expression
1 ffn 6720 . 2 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 df-fn 6549 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3 eqimss 4037 . . . . 5 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
4 cnvi 6145 . . . . . . . . . 10 I = I
54reseq1i 5977 . . . . . . . . 9 ( I ↾ 𝐴) = ( I ↾ 𝐴)
65cnveqi 5873 . . . . . . . 8 ( I ↾ 𝐴) = ( I ↾ 𝐴)
7 cnvresid 6630 . . . . . . . 8 ( I ↾ 𝐴) = ( I ↾ 𝐴)
86, 7eqtr2i 2755 . . . . . . 7 ( I ↾ 𝐴) = ( I ↾ 𝐴)
98coeq2i 5859 . . . . . 6 (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹( I ↾ 𝐴))
10 cores2 6262 . . . . . 6 (dom 𝐹𝐴 → (𝐹( I ↾ 𝐴)) = (𝐹 ∘ I ))
119, 10eqtrid 2778 . . . . 5 (dom 𝐹𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
123, 11syl 17 . . . 4 (dom 𝐹 = 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
13 funrel 6568 . . . . 5 (Fun 𝐹 → Rel 𝐹)
14 coi1 6265 . . . . 5 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
1513, 14syl 17 . . . 4 (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
1612, 15sylan9eqr 2788 . . 3 ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
172, 16sylbi 216 . 2 (𝐹 Fn 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
181, 17syl 17 1 (𝐹:𝐴𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wss 3946   I cid 5571  ccnv 5673  dom cdm 5674  cres 5676  ccom 5678  Rel wrel 5679  Fun wfun 6540   Fn wfn 6541  wf 6542
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-10 2130  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
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-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-fun 6548  df-fn 6549  df-f 6550
This theorem is referenced by:  fcof1oinvd  7299  mapen  9171  mapfien  9444  hashfacen  14466  hashfacenOLD  14467  cofurid  17905  setccatid  18101  estrccatid  18150  curf2ndf  18267  efmndid  18873  efmndmnd  18874  f1omvdco2  19442  psgnunilem1  19487  pf1mpf  22340  pf1ind  22343  wilthlem3  27095  hoico1  31686  fmptco1f1o  32550  fcobijfs  32637  cycpmconjslem2  33037  cycpmconjs  33038  cyc3conja  33039  1arithidomlem2  33417  reprpmtf1o  34485  ltrncoidN  39840  trlcoabs2N  40434  trlcoat  40435  cdlemg47a  40446  cdlemg46  40447  trljco  40452  tendo1mulr  40483  tendo0co2  40500  cdlemi2  40531  cdlemk2  40544  cdlemk4  40546  cdlemk8  40550  cdlemk53  40669  cdlemk55a  40671  dvhopN  40828  dihopelvalcpre  40960  dihmeetlem1N  41002  dihglblem5apreN  41003  diophrw  42453  mendring  42890  rngccatidALTV  47685  ringccatidALTV  47719
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