![]() |
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 6501 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | cores 6202 | . . 3 ⊢ (ran 𝐹 ⊆ 𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹)) | |
3 | fnrel 6605 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
4 | coi2 6216 | . . . 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 216 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ⊆ wss 3911 I cid 5531 ran crn 5635 ↾ cres 5636 ∘ ccom 5638 Rel wrel 5639 Fn wfn 6492 ⟶wf 6493 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-fun 6499 df-fn 6500 df-f 6501 |
This theorem is referenced by: fcof1oinvd 7240 mapen 9086 mapfien 9345 hashfacen 14352 hashfacenOLD 14353 cofulid 17777 setccatid 17971 estrccatid 18020 efmndid 18699 efmndmnd 18700 symggrp 19183 f1omvdco2 19231 symggen 19253 psgnunilem1 19276 gsumval3 19685 gsumzf1o 19690 frgpcyg 20983 f1linds 21234 qtophmeo 23171 motgrp 27488 hoico2 30702 fcoinver 31528 fcobij 31642 symgfcoeu 31936 symgcom 31937 pmtrcnel2 31944 cycpmconjs 32008 subfacp1lem5 33781 ltrncoidN 38594 trlcoat 39189 trlcone 39194 cdlemg47a 39200 cdlemg47 39202 trljco 39206 tgrpgrplem 39215 tendo1mul 39236 tendo0pl 39257 cdlemkid2 39390 cdlemk45 39413 cdlemk53b 39422 erng1r 39461 tendocnv 39487 dvalveclem 39491 dva0g 39493 dvhgrp 39573 dvhlveclem 39574 dvh0g 39577 cdlemn8 39670 dihordlem7b 39681 dihopelvalcpre 39714 mendring 41522 rngccatidALTV 46294 ringccatidALTV 46357 |
Copyright terms: Public domain | W3C validator |