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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 6736 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
| 2 | df-fn 6564 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
| 3 | eqimss 4042 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
| 4 | cnvi 6161 | . . . . . . . . . 10 ⊢ ◡ I = I | |
| 5 | 4 | reseq1i 5993 | . . . . . . . . 9 ⊢ (◡ I ↾ 𝐴) = ( I ↾ 𝐴) |
| 6 | 5 | cnveqi 5885 | . . . . . . . 8 ⊢ ◡(◡ I ↾ 𝐴) = ◡( I ↾ 𝐴) |
| 7 | cnvresid 6645 | . . . . . . . 8 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) | |
| 8 | 6, 7 | eqtr2i 2766 | . . . . . . 7 ⊢ ( I ↾ 𝐴) = ◡(◡ I ↾ 𝐴) |
| 9 | 8 | coeq2i 5871 | . . . . . 6 ⊢ (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ ◡(◡ I ↾ 𝐴)) |
| 10 | cores2 6279 | . . . . . 6 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ◡(◡ I ↾ 𝐴)) = (𝐹 ∘ I )) | |
| 11 | 9, 10 | eqtrid 2789 | . . . . 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 6583 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | |
| 14 | coi1 6282 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹) | |
| 15 | 13, 14 | syl 17 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹) |
| 16 | 12, 15 | sylan9eqr 2799 | . . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
| 17 | 2, 16 | sylbi 217 | . 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 395 = wceq 1540 ⊆ wss 3951 I cid 5577 ◡ccnv 5684 dom cdm 5685 ↾ cres 5687 ∘ ccom 5689 Rel wrel 5690 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: fcof1oinvd 7313 mapen 9181 mapfien 9448 hashfacen 14493 cofurid 17936 setccatid 18129 estrccatid 18176 curf2ndf 18292 efmndid 18901 efmndmnd 18902 f1omvdco2 19466 psgnunilem1 19511 pf1mpf 22356 pf1ind 22359 wilthlem3 27113 hoico1 31775 fmptco1f1o 32643 fcobijfs 32734 cycpmconjslem2 33175 cycpmconjs 33176 cyc3conja 33177 1arithidomlem2 33564 reprpmtf1o 34641 ltrncoidN 40130 trlcoabs2N 40724 trlcoat 40725 cdlemg47a 40736 cdlemg46 40737 trljco 40742 tendo1mulr 40773 tendo0co2 40790 cdlemi2 40821 cdlemk2 40834 cdlemk4 40836 cdlemk8 40840 cdlemk53 40959 cdlemk55a 40961 dvhopN 41118 dihopelvalcpre 41250 dihmeetlem1N 41292 dihglblem5apreN 41293 diophrw 42770 mendring 43200 rngccatidALTV 48188 ringccatidALTV 48222 |
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