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Theorem fcoi1 4995
 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 4972 . 2 (𝐹:AB𝐹 Fn A)
2 df-fn 4832 . . 3 (𝐹 Fn A ↔ (Fun 𝐹 dom 𝐹 = A))
3 eqimss 2974 . . . . 5 (dom 𝐹 = A → dom 𝐹A)
4 cnvi 4655 . . . . . . . . . 10 I = I
54reseq1i 4535 . . . . . . . . 9 ( I ↾ A) = ( I ↾ A)
65cnveqi 4437 . . . . . . . 8 ( I ↾ A) = ( I ↾ A)
7 cnvresid 4899 . . . . . . . 8 ( I ↾ A) = ( I ↾ A)
86, 7eqtr2i 2043 . . . . . . 7 ( I ↾ A) = ( I ↾ A)
98coeq2i 4423 . . . . . 6 (𝐹 ∘ ( I ↾ A)) = (𝐹( I ↾ A))
10 cores2 4760 . . . . . 6 (dom 𝐹A → (𝐹( I ↾ A)) = (𝐹 ∘ I ))
119, 10syl5eq 2066 . . . . 5 (dom 𝐹A → (𝐹 ∘ ( I ↾ A)) = (𝐹 ∘ I ))
123, 11syl 14 . . . 4 (dom 𝐹 = A → (𝐹 ∘ ( I ↾ A)) = (𝐹 ∘ I ))
13 funrel 4845 . . . . 5 (Fun 𝐹 → Rel 𝐹)
14 coi1 4763 . . . . 5 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
1513, 14syl 14 . . . 4 (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
1612, 15sylan9eqr 2076 . . 3 ((Fun 𝐹 dom 𝐹 = A) → (𝐹 ∘ ( I ↾ A)) = 𝐹)
172, 16sylbi 114 . 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 97   = wceq 1228   ⊆ wss 2894   I cid 3999  ◡ccnv 4271  dom cdm 4272   ↾ cres 4274   ∘ ccom 4276  Rel wrel 4277  Fun wfun 4823   Fn wfn 4824  ⟶wf 4825 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-fun 4831  df-fn 4832  df-f 4833 This theorem is referenced by:  fcof1o  5354
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