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Theorem fcoi1 5239
 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 5208 . 2 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 df-fn 5062 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3 eqimss 3101 . . . . 5 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
4 cnvi 4879 . . . . . . . . . 10 I = I
54reseq1i 4751 . . . . . . . . 9 ( I ↾ 𝐴) = ( I ↾ 𝐴)
65cnveqi 4652 . . . . . . . 8 ( I ↾ 𝐴) = ( I ↾ 𝐴)
7 cnvresid 5133 . . . . . . . 8 ( I ↾ 𝐴) = ( I ↾ 𝐴)
86, 7eqtr2i 2121 . . . . . . 7 ( I ↾ 𝐴) = ( I ↾ 𝐴)
98coeq2i 4637 . . . . . 6 (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹( I ↾ 𝐴))
10 cores2 4987 . . . . . 6 (dom 𝐹𝐴 → (𝐹( I ↾ 𝐴)) = (𝐹 ∘ I ))
119, 10syl5eq 2144 . . . . 5 (dom 𝐹𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
123, 11syl 14 . . . 4 (dom 𝐹 = 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
13 funrel 5076 . . . . 5 (Fun 𝐹 → Rel 𝐹)
14 coi1 4990 . . . . 5 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
1513, 14syl 14 . . . 4 (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
1612, 15sylan9eqr 2154 . . 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 1299   ⊆ wss 3021   I cid 4148  ◡ccnv 4476  dom cdm 4477   ↾ cres 4479   ∘ ccom 4481  Rel wrel 4482  Fun wfun 5053   Fn wfn 5054  ⟶wf 5055 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069 This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-fun 5061  df-fn 5062  df-f 5063 This theorem is referenced by:  fcof1o  5622  mapen  6669  hashfacen  10420
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