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Theorem fcoi1 6239
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 6206 . 2 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 df-fn 6052 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3 eqimss 3798 . . . . 5 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
4 cnvi 5695 . . . . . . . . . 10 I = I
54reseq1i 5547 . . . . . . . . 9 ( I ↾ 𝐴) = ( I ↾ 𝐴)
65cnveqi 5452 . . . . . . . 8 ( I ↾ 𝐴) = ( I ↾ 𝐴)
7 cnvresid 6129 . . . . . . . 8 ( I ↾ 𝐴) = ( I ↾ 𝐴)
86, 7eqtr2i 2783 . . . . . . 7 ( I ↾ 𝐴) = ( I ↾ 𝐴)
98coeq2i 5438 . . . . . 6 (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹( I ↾ 𝐴))
10 cores2 5809 . . . . . 6 (dom 𝐹𝐴 → (𝐹( I ↾ 𝐴)) = (𝐹 ∘ I ))
119, 10syl5eq 2806 . . . . 5 (dom 𝐹𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
123, 11syl 17 . . . 4 (dom 𝐹 = 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
13 funrel 6066 . . . . 5 (Fun 𝐹 → Rel 𝐹)
14 coi1 5812 . . . . 5 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
1513, 14syl 17 . . . 4 (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
1612, 15sylan9eqr 2816 . . 3 ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
172, 16sylbi 207 . 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 383   = wceq 1632  wss 3715   I cid 5173  ccnv 5265  dom cdm 5266  cres 5268  ccom 5270  Rel wrel 5271  Fun wfun 6043   Fn wfn 6044  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-fun 6051  df-fn 6052  df-f 6053
This theorem is referenced by:  fcof1oinvd  6711  mapen  8289  mapfien  8478  hashfacen  13430  cofurid  16752  setccatid  16935  estrccatid  16973  curf2ndf  17088  symgid  18021  f1omvdco2  18068  psgnunilem1  18113  pf1mpf  19918  pf1ind  19921  wilthlem3  24995  hoico1  28924  fmptco1f1o  29743  fcobijfs  29810  reprpmtf1o  31013  ltrncoidN  35917  trlcoabs2N  36512  trlcoat  36513  cdlemg47a  36524  cdlemg46  36525  trljco  36530  tendo1mulr  36561  tendo0co2  36578  cdlemi2  36609  cdlemk2  36622  cdlemk4  36624  cdlemk8  36628  cdlemk53  36747  cdlemk55a  36749  dvhopN  36907  dihopelvalcpre  37039  dihmeetlem1N  37081  dihglblem5apreN  37082  diophrw  37824  mendring  38264  rngccatidALTV  42499  ringccatidALTV  42562
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