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Theorem fcoi2 5176
Description: Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fcoi2 (𝐹:𝐴𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)

Proof of Theorem fcoi2
StepHypRef Expression
1 df-f 5006 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 cores 4921 . . 3 (ran 𝐹𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹))
3 fnrel 5098 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
4 coi2 4934 . . . 4 (Rel 𝐹 → ( I ∘ 𝐹) = 𝐹)
53, 4syl 14 . . 3 (𝐹 Fn 𝐴 → ( I ∘ 𝐹) = 𝐹)
62, 5sylan9eqr 2142 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
71, 6sylbi 119 1 (𝐹:𝐴𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wss 2997   I cid 4106  ran crn 4429  cres 4430  ccom 4432  Rel wrel 4433   Fn wfn 4997  wf 4998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-fun 5004  df-fn 5005  df-f 5006
This theorem is referenced by:  fcof1o  5550  mapen  6542  hashfacen  10206
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