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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 5331 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | df-fn 5185 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
3 | eqimss 3191 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
4 | cnvi 5002 | . . . . . . . . . 10 ⊢ ◡ I = I | |
5 | 4 | reseq1i 4874 | . . . . . . . . 9 ⊢ (◡ I ↾ 𝐴) = ( I ↾ 𝐴) |
6 | 5 | cnveqi 4773 | . . . . . . . 8 ⊢ ◡(◡ I ↾ 𝐴) = ◡( I ↾ 𝐴) |
7 | cnvresid 5256 | . . . . . . . 8 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) | |
8 | 6, 7 | eqtr2i 2186 | . . . . . . 7 ⊢ ( I ↾ 𝐴) = ◡(◡ I ↾ 𝐴) |
9 | 8 | coeq2i 4758 | . . . . . 6 ⊢ (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ ◡(◡ I ↾ 𝐴)) |
10 | cores2 5110 | . . . . . 6 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ◡(◡ I ↾ 𝐴)) = (𝐹 ∘ I )) | |
11 | 9, 10 | syl5eq 2209 | . . . . 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 5199 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | |
14 | coi1 5113 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹) | |
15 | 13, 14 | syl 14 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹) |
16 | 12, 15 | sylan9eqr 2219 | . . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
17 | 2, 16 | sylbi 120 | . 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 103 = wceq 1342 ⊆ wss 3111 I cid 4260 ◡ccnv 4597 dom cdm 4598 ↾ cres 4600 ∘ ccom 4602 Rel wrel 4603 Fun wfun 5176 Fn wfn 5177 ⟶wf 5178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-fun 5184 df-fn 5185 df-f 5186 |
This theorem is referenced by: fcof1o 5751 mapen 6803 hashfacen 10735 |
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