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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 5930 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | cores 5676 | . . 3 ⊢ (ran 𝐹 ⊆ 𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹)) | |
3 | fnrel 6027 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
4 | coi2 5690 | . . . 4 ⊢ (Rel 𝐹 → ( I ∘ 𝐹) = 𝐹) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → ( I ∘ 𝐹) = 𝐹) |
6 | 2, 5 | sylan9eqr 2707 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹) |
7 | 1, 6 | sylbi 207 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ⊆ wss 3607 I cid 5052 ran crn 5144 ↾ cres 5145 ∘ ccom 5147 Rel wrel 5148 Fn wfn 5921 ⟶wf 5922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-fun 5928 df-fn 5929 df-f 5930 |
This theorem is referenced by: fcof1oinvd 6588 mapen 8165 mapfien 8354 hashfacen 13276 cofulid 16597 setccatid 16781 estrccatid 16819 symggrp 17866 f1omvdco2 17914 symggen 17936 psgnunilem1 17959 gsumval3 18354 gsumzf1o 18359 frgpcyg 19970 f1linds 20212 qtophmeo 21668 motgrp 25483 hoico2 28744 fcoinver 29544 fcobij 29628 symgfcoeu 29973 subfacp1lem5 31292 ltrncoidN 35732 trlcoat 36328 trlcone 36333 cdlemg47a 36339 cdlemg47 36341 trljco 36345 tgrpgrplem 36354 tendo1mul 36375 tendo0pl 36396 cdlemkid2 36529 cdlemk45 36552 cdlemk53b 36561 erng1r 36600 tendocnv 36627 dvalveclem 36631 dva0g 36633 dvhgrp 36713 dvhlveclem 36714 dvh0g 36717 cdlemn8 36810 dihordlem7b 36821 dihopelvalcpre 36854 mendring 38079 rngccatidALTV 42314 ringccatidALTV 42377 |
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