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Theorem fcoi1 5013
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 4989 . 2 (𝐹:AB𝐹 Fn A)
2 df-fn 4848 . . 3 (𝐹 Fn A ↔ (Fun 𝐹 dom 𝐹 = A))
3 eqimss 2991 . . . . 5 (dom 𝐹 = A → dom 𝐹A)
4 cnvi 4671 . . . . . . . . . 10 I = I
54reseq1i 4551 . . . . . . . . 9 ( I ↾ A) = ( I ↾ A)
65cnveqi 4453 . . . . . . . 8 ( I ↾ A) = ( I ↾ A)
7 cnvresid 4916 . . . . . . . 8 ( I ↾ A) = ( I ↾ A)
86, 7eqtr2i 2058 . . . . . . 7 ( I ↾ A) = ( I ↾ A)
98coeq2i 4439 . . . . . 6 (𝐹 ∘ ( I ↾ A)) = (𝐹( I ↾ A))
10 cores2 4776 . . . . . 6 (dom 𝐹A → (𝐹( I ↾ A)) = (𝐹 ∘ I ))
119, 10syl5eq 2081 . . . . 5 (dom 𝐹A → (𝐹 ∘ ( I ↾ A)) = (𝐹 ∘ I ))
123, 11syl 14 . . . 4 (dom 𝐹 = A → (𝐹 ∘ ( I ↾ A)) = (𝐹 ∘ I ))
13 funrel 4862 . . . . 5 (Fun 𝐹 → Rel 𝐹)
14 coi1 4779 . . . . 5 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
1513, 14syl 14 . . . 4 (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
1612, 15sylan9eqr 2091 . . 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 1242  wss 2911   I cid 4016  ccnv 4287  dom cdm 4288  cres 4290  ccom 4292  Rel wrel 4293  Fun wfun 4839   Fn wfn 4840  wf 4841
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847  df-fn 4848  df-f 4849
This theorem is referenced by:  fcof1o  5372
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