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Mirrors > Home > ILE Home > Th. List > fcoi1 | GIF version |
Description: Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fcoi1 | ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5046 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | df-fn 4905 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
3 | eqimss 2997 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
4 | cnvi 4728 | . . . . . . . . . 10 ⊢ ◡ I = I | |
5 | 4 | reseq1i 4608 | . . . . . . . . 9 ⊢ (◡ I ↾ 𝐴) = ( I ↾ 𝐴) |
6 | 5 | cnveqi 4510 | . . . . . . . 8 ⊢ ◡(◡ I ↾ 𝐴) = ◡( I ↾ 𝐴) |
7 | cnvresid 4973 | . . . . . . . 8 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) | |
8 | 6, 7 | eqtr2i 2061 | . . . . . . 7 ⊢ ( I ↾ 𝐴) = ◡(◡ I ↾ 𝐴) |
9 | 8 | coeq2i 4496 | . . . . . 6 ⊢ (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ ◡(◡ I ↾ 𝐴)) |
10 | cores2 4833 | . . . . . 6 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ◡(◡ I ↾ 𝐴)) = (𝐹 ∘ I )) | |
11 | 9, 10 | syl5eq 2084 | . . . . 5 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I )) |
12 | 3, 11 | syl 14 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I )) |
13 | funrel 4919 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | |
14 | coi1 4836 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹) | |
15 | 13, 14 | syl 14 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹) |
16 | 12, 15 | sylan9eqr 2094 | . . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
17 | 2, 16 | sylbi 114 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
18 | 1, 17 | syl 14 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ⊆ wss 2917 I cid 4025 ◡ccnv 4344 dom cdm 4345 ↾ cres 4347 ∘ ccom 4349 Rel wrel 4350 Fun wfun 4896 Fn wfn 4897 ⟶wf 4898 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-fun 4904 df-fn 4905 df-f 4906 |
This theorem is referenced by: fcof1o 5429 |
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