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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 6558 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | cores 6260 | . . 3 ⊢ (ran 𝐹 ⊆ 𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹)) | |
3 | fnrel 6662 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
4 | coi2 6274 | . . . 4 ⊢ (Rel 𝐹 → ( I ∘ 𝐹) = 𝐹) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → ( I ∘ 𝐹) = 𝐹) |
6 | 2, 5 | sylan9eqr 2788 | . 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 394 = wceq 1534 ⊆ wss 3947 I cid 5579 ran crn 5683 ↾ cres 5684 ∘ ccom 5686 Rel wrel 5687 Fn wfn 6549 ⟶wf 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-fun 6556 df-fn 6557 df-f 6558 |
This theorem is referenced by: fcof1oinvd 7307 mapen 9179 mapfien 9451 hashfacen 14471 hashfacenOLD 14472 cofulid 17909 setccatid 18106 estrccatid 18155 efmndid 18878 efmndmnd 18879 symggrp 19398 f1omvdco2 19446 symggen 19468 psgnunilem1 19491 gsumval3 19905 gsumzf1o 19910 frgpcyg 21571 f1linds 21823 qtophmeo 23812 motgrp 28470 hoico2 31690 fcoinver 32524 fcobij 32636 symgfcoeu 32960 symgcom 32961 pmtrcnel2 32968 cycpmconjs 33034 subfacp1lem5 35012 ltrncoidN 39827 trlcoat 40422 trlcone 40427 cdlemg47a 40433 cdlemg47 40435 trljco 40439 tgrpgrplem 40448 tendo1mul 40469 tendo0pl 40490 cdlemkid2 40623 cdlemk45 40646 cdlemk53b 40655 erng1r 40694 tendocnv 40720 dvalveclem 40724 dva0g 40726 dvhgrp 40806 dvhlveclem 40807 dvh0g 40810 cdlemn8 40903 dihordlem7b 40914 dihopelvalcpre 40947 aks6d1c6lem5 41875 mendring 42853 rngccatidALTV 47649 ringccatidALTV 47683 |
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