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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 6566 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | cores 6270 | . . 3 ⊢ (ran 𝐹 ⊆ 𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹)) | |
3 | fnrel 6670 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
4 | coi2 6284 | . . . 4 ⊢ (Rel 𝐹 → ( I ∘ 𝐹) = 𝐹) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → ( I ∘ 𝐹) = 𝐹) |
6 | 2, 5 | sylan9eqr 2796 | . 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 1536 ⊆ wss 3962 I cid 5581 ran crn 5689 ↾ cres 5690 ∘ ccom 5692 Rel wrel 5693 Fn wfn 6557 ⟶wf 6558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-fun 6564 df-fn 6565 df-f 6566 |
This theorem is referenced by: fcof1oinvd 7312 mapen 9179 mapfien 9445 hashfacen 14489 cofulid 17940 setccatid 18137 estrccatid 18186 efmndid 18913 efmndmnd 18914 symggrp 19432 f1omvdco2 19480 symggen 19502 psgnunilem1 19525 gsumval3 19939 gsumzf1o 19944 frgpcyg 21609 f1linds 21862 qtophmeo 23840 motgrp 28565 hoico2 31785 fcoinver 32623 fcobij 32739 symgfcoeu 33084 symgcom 33085 pmtrcnel2 33092 cycpmconjs 33158 subfacp1lem5 35168 ltrncoidN 40110 trlcoat 40705 trlcone 40710 cdlemg47a 40716 cdlemg47 40718 trljco 40722 tgrpgrplem 40731 tendo1mul 40752 tendo0pl 40773 cdlemkid2 40906 cdlemk45 40929 cdlemk53b 40938 erng1r 40977 tendocnv 41003 dvalveclem 41007 dva0g 41009 dvhgrp 41089 dvhlveclem 41090 dvh0g 41093 cdlemn8 41186 dihordlem7b 41197 dihopelvalcpre 41230 aks6d1c6lem5 42158 mendring 43176 rngccatidALTV 48115 ringccatidALTV 48149 |
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