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Theorem fcoi1 5273
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 5242 . 2 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 df-fn 5096 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3 eqimss 3121 . . . . 5 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
4 cnvi 4913 . . . . . . . . . 10 I = I
54reseq1i 4785 . . . . . . . . 9 ( I ↾ 𝐴) = ( I ↾ 𝐴)
65cnveqi 4684 . . . . . . . 8 ( I ↾ 𝐴) = ( I ↾ 𝐴)
7 cnvresid 5167 . . . . . . . 8 ( I ↾ 𝐴) = ( I ↾ 𝐴)
86, 7eqtr2i 2139 . . . . . . 7 ( I ↾ 𝐴) = ( I ↾ 𝐴)
98coeq2i 4669 . . . . . 6 (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹( I ↾ 𝐴))
10 cores2 5021 . . . . . 6 (dom 𝐹𝐴 → (𝐹( I ↾ 𝐴)) = (𝐹 ∘ I ))
119, 10syl5eq 2162 . . . . 5 (dom 𝐹𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
123, 11syl 14 . . . 4 (dom 𝐹 = 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
13 funrel 5110 . . . . 5 (Fun 𝐹 → Rel 𝐹)
14 coi1 5024 . . . . 5 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
1513, 14syl 14 . . . 4 (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
1612, 15sylan9eqr 2172 . . 3 ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
172, 16sylbi 120 . 2 (𝐹 Fn 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
181, 17syl 14 1 (𝐹:𝐴𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wss 3041   I cid 4180  ccnv 4508  dom cdm 4509  cres 4511  ccom 4513  Rel wrel 4514  Fun wfun 5087   Fn wfn 5088  wf 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-fun 5095  df-fn 5096  df-f 5097
This theorem is referenced by:  fcof1o  5658  mapen  6708  hashfacen  10547
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