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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 6548 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | cores 6249 | . . 3 ⊢ (ran 𝐹 ⊆ 𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹)) | |
3 | fnrel 6652 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
4 | coi2 6263 | . . . 4 ⊢ (Rel 𝐹 → ( I ∘ 𝐹) = 𝐹) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → ( I ∘ 𝐹) = 𝐹) |
6 | 2, 5 | sylan9eqr 2795 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹) |
7 | 1, 6 | sylbi 216 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ⊆ wss 3949 I cid 5574 ran crn 5678 ↾ cres 5679 ∘ ccom 5681 Rel wrel 5682 Fn wfn 6539 ⟶wf 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-fun 6546 df-fn 6547 df-f 6548 |
This theorem is referenced by: fcof1oinvd 7291 mapen 9141 mapfien 9403 hashfacen 14413 hashfacenOLD 14414 cofulid 17840 setccatid 18034 estrccatid 18083 efmndid 18769 efmndmnd 18770 symggrp 19268 f1omvdco2 19316 symggen 19338 psgnunilem1 19361 gsumval3 19775 gsumzf1o 19780 frgpcyg 21129 f1linds 21380 qtophmeo 23321 motgrp 27794 hoico2 31010 fcoinver 31835 fcobij 31947 symgfcoeu 32243 symgcom 32244 pmtrcnel2 32251 cycpmconjs 32315 subfacp1lem5 34175 ltrncoidN 38999 trlcoat 39594 trlcone 39599 cdlemg47a 39605 cdlemg47 39607 trljco 39611 tgrpgrplem 39620 tendo1mul 39641 tendo0pl 39662 cdlemkid2 39795 cdlemk45 39818 cdlemk53b 39827 erng1r 39866 tendocnv 39892 dvalveclem 39896 dva0g 39898 dvhgrp 39978 dvhlveclem 39979 dvh0g 39982 cdlemn8 40075 dihordlem7b 40086 dihopelvalcpre 40119 mendring 41934 rngccatidALTV 46887 ringccatidALTV 46950 |
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