| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > fcoi2 | Structured version Visualization version GIF version | ||
| Description: Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
| Ref | Expression |
|---|---|
| fcoi2 | ⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-f 6565 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
| 2 | cores 6269 | . . 3 ⊢ (ran 𝐹 ⊆ 𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹)) | |
| 3 | fnrel 6670 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
| 4 | coi2 6283 | . . . 4 ⊢ (Rel 𝐹 → ( I ∘ 𝐹) = 𝐹) | |
| 5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → ( I ∘ 𝐹) = 𝐹) |
| 6 | 2, 5 | sylan9eqr 2799 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹) |
| 7 | 1, 6 | sylbi 217 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ⊆ wss 3951 I cid 5577 ran crn 5686 ↾ cres 5687 ∘ ccom 5689 Rel wrel 5690 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-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-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-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: fcof1oinvd 7313 mapen 9181 mapfien 9448 hashfacen 14493 cofulid 17935 setccatid 18129 estrccatid 18176 efmndid 18901 efmndmnd 18902 symggrp 19418 f1omvdco2 19466 symggen 19488 psgnunilem1 19511 gsumval3 19925 gsumzf1o 19930 frgpcyg 21592 f1linds 21845 qtophmeo 23825 motgrp 28551 hoico2 31776 fcoinver 32617 fcobij 32733 symgfcoeu 33102 symgcom 33103 pmtrcnel2 33110 cycpmconjs 33176 subfacp1lem5 35189 ltrncoidN 40130 trlcoat 40725 trlcone 40730 cdlemg47a 40736 cdlemg47 40738 trljco 40742 tgrpgrplem 40751 tendo1mul 40772 tendo0pl 40793 cdlemkid2 40926 cdlemk45 40949 cdlemk53b 40958 erng1r 40997 tendocnv 41023 dvalveclem 41027 dva0g 41029 dvhgrp 41109 dvhlveclem 41110 dvh0g 41113 cdlemn8 41206 dihordlem7b 41217 dihopelvalcpre 41250 aks6d1c6lem5 42178 mendring 43200 rngccatidALTV 48188 ringccatidALTV 48222 |
| Copyright terms: Public domain | W3C validator |