Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6514 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | df-fn 6358 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
3 | eqimss 4023 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
4 | cnvi 6000 | . . . . . . . . . 10 ⊢ ◡ I = I | |
5 | 4 | reseq1i 5849 | . . . . . . . . 9 ⊢ (◡ I ↾ 𝐴) = ( I ↾ 𝐴) |
6 | 5 | cnveqi 5745 | . . . . . . . 8 ⊢ ◡(◡ I ↾ 𝐴) = ◡( I ↾ 𝐴) |
7 | cnvresid 6433 | . . . . . . . 8 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) | |
8 | 6, 7 | eqtr2i 2845 | . . . . . . 7 ⊢ ( I ↾ 𝐴) = ◡(◡ I ↾ 𝐴) |
9 | 8 | coeq2i 5731 | . . . . . 6 ⊢ (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ ◡(◡ I ↾ 𝐴)) |
10 | cores2 6112 | . . . . . 6 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ◡(◡ I ↾ 𝐴)) = (𝐹 ∘ I )) | |
11 | 9, 10 | syl5eq 2868 | . . . . 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 6372 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | |
14 | coi1 6115 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹) |
16 | 12, 15 | sylan9eqr 2878 | . . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
17 | 2, 16 | sylbi 219 | . 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 398 = wceq 1537 ⊆ wss 3936 I cid 5459 ◡ccnv 5554 dom cdm 5555 ↾ cres 5557 ∘ ccom 5559 Rel wrel 5560 Fun wfun 6349 Fn wfn 6350 ⟶wf 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-fun 6357 df-fn 6358 df-f 6359 |
This theorem is referenced by: fcof1oinvd 7049 mapen 8681 mapfien 8871 hashfacen 13813 cofurid 17161 setccatid 17344 estrccatid 17382 curf2ndf 17497 efmndid 18053 efmndmnd 18054 f1omvdco2 18576 psgnunilem1 18621 pf1mpf 20515 pf1ind 20518 wilthlem3 25647 hoico1 29533 fmptco1f1o 30378 fcobijfs 30459 cycpmconjslem2 30797 cycpmconjs 30798 cyc3conja 30799 reprpmtf1o 31897 ltrncoidN 37279 trlcoabs2N 37873 trlcoat 37874 cdlemg47a 37885 cdlemg46 37886 trljco 37891 tendo1mulr 37922 tendo0co2 37939 cdlemi2 37970 cdlemk2 37983 cdlemk4 37985 cdlemk8 37989 cdlemk53 38108 cdlemk55a 38110 dvhopN 38267 dihopelvalcpre 38399 dihmeetlem1N 38441 dihglblem5apreN 38442 diophrw 39376 mendring 39812 rngccatidALTV 44280 ringccatidALTV 44343 |
Copyright terms: Public domain | W3C validator |