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Mirrors > Home > MPE Home > Th. List > fcoi1 | Structured version Visualization version 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 5958 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | df-fn 5807 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
3 | eqimss 3620 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
4 | cnvi 5456 | . . . . . . . . . 10 ⊢ ◡ I = I | |
5 | 4 | reseq1i 5313 | . . . . . . . . 9 ⊢ (◡ I ↾ 𝐴) = ( I ↾ 𝐴) |
6 | 5 | cnveqi 5219 | . . . . . . . 8 ⊢ ◡(◡ I ↾ 𝐴) = ◡( I ↾ 𝐴) |
7 | cnvresid 5882 | . . . . . . . 8 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) | |
8 | 6, 7 | eqtr2i 2633 | . . . . . . 7 ⊢ ( I ↾ 𝐴) = ◡(◡ I ↾ 𝐴) |
9 | 8 | coeq2i 5204 | . . . . . 6 ⊢ (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ ◡(◡ I ↾ 𝐴)) |
10 | cores2 5565 | . . . . . 6 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ◡(◡ I ↾ 𝐴)) = (𝐹 ∘ I )) | |
11 | 9, 10 | syl5eq 2656 | . . . . 5 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I )) |
12 | 3, 11 | syl 17 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I )) |
13 | funrel 5821 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | |
14 | coi1 5568 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹) |
16 | 12, 15 | sylan9eqr 2666 | . . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
17 | 2, 16 | sylbi 206 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
18 | 1, 17 | syl 17 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ⊆ wss 3540 I cid 4948 ◡ccnv 5037 dom cdm 5038 ↾ cres 5040 ∘ ccom 5042 Rel wrel 5043 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 |
This theorem is referenced by: fcof1oinvd 6448 mapen 8009 mapfien 8196 hashfacen 13095 cofurid 16374 setccatid 16557 estrccatid 16595 curf2ndf 16710 symgid 17644 f1omvdco2 17691 psgnunilem1 17736 pf1mpf 19537 pf1ind 19540 wilthlem3 24596 hoico1 27999 fcobijfs 28889 ltrncoidN 34432 trlcoabs2N 35028 trlcoat 35029 cdlemg47a 35040 cdlemg46 35041 trljco 35046 tendo1mulr 35077 tendo0co2 35094 cdlemi2 35125 cdlemk2 35138 cdlemk4 35140 cdlemk8 35144 cdlemk53 35263 cdlemk55a 35265 dvhopN 35423 dihopelvalcpre 35555 dihmeetlem1N 35597 dihglblem5apreN 35598 diophrw 36340 mendring 36781 rngccatidALTV 41781 ringccatidALTV 41844 |
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