ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fcoi1 GIF version

Theorem fcoi1 5098
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 5074 . 2 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 df-fn 4933 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3 eqimss 3025 . . . . 5 (dom 𝐹 = 𝐴 → dom 𝐹𝐴)
4 cnvi 4756 . . . . . . . . . 10 I = I
54reseq1i 4636 . . . . . . . . 9 ( I ↾ 𝐴) = ( I ↾ 𝐴)
65cnveqi 4538 . . . . . . . 8 ( I ↾ 𝐴) = ( I ↾ 𝐴)
7 cnvresid 5001 . . . . . . . 8 ( I ↾ 𝐴) = ( I ↾ 𝐴)
86, 7eqtr2i 2077 . . . . . . 7 ( I ↾ 𝐴) = ( I ↾ 𝐴)
98coeq2i 4524 . . . . . 6 (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹( I ↾ 𝐴))
10 cores2 4861 . . . . . 6 (dom 𝐹𝐴 → (𝐹( I ↾ 𝐴)) = (𝐹 ∘ I ))
119, 10syl5eq 2100 . . . . 5 (dom 𝐹𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
123, 11syl 14 . . . 4 (dom 𝐹 = 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
13 funrel 4947 . . . . 5 (Fun 𝐹 → Rel 𝐹)
14 coi1 4864 . . . . 5 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
1513, 14syl 14 . . . 4 (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
1612, 15sylan9eqr 2110 . . 3 ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
172, 16sylbi 118 . 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 101   = wceq 1259  wss 2945   I cid 4053  ccnv 4372  dom cdm 4373  cres 4375  ccom 4377  Rel wrel 4378  Fun wfun 4924   Fn wfn 4925  wf 4926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-fun 4932  df-fn 4933  df-f 4934
This theorem is referenced by:  fcof1o  5457
  Copyright terms: Public domain W3C validator