Theorem fcoi2 6703

Description: Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fcoi2	⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)

Proof of Theorem fcoi2

Step	Hyp	Ref	Expression
1		df-f 6490	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		cores 6202	. . 3 ⊢ (ran 𝐹 ⊆ 𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹))
3		fnrel 6588	. . . 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 2786	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
7	1, 6	sylbi 217	1 ⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ⊆ wss 3905   I cid 5517  ran crn 5624   ↾ cres 5625   ∘ ccom 5627  Rel wrel 5628   Fn wfn 6481  ⟶wf 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-fun 6488  df-fn 6489  df-f 6490

This theorem is referenced by:  fcof1oinvd  7234  mapen  9065  mapfien  9317  hashfacen  14379  cofulid  17815  setccatid  18009  estrccatid  18056  efmndid  18780  efmndmnd  18781  symggrp  19297  f1omvdco2  19345  symggen  19367  psgnunilem1  19390  gsumval3  19804  gsumzf1o  19809  frgpcyg  21498  f1linds  21750  qtophmeo  23720  motgrp  28506  hoico2  31719  fcoinver  32566  fcobij  32678  symgfcoeu  33037  symgcom  33038  pmtrcnel2  33045  cycpmconjs  33111  subfacp1lem5  35156  ltrncoidN  40107  trlcoat  40702  trlcone  40707  cdlemg47a  40713  cdlemg47  40715  trljco  40719  tgrpgrplem  40728  tendo1mul  40749  tendo0pl  40770  cdlemkid2  40903  cdlemk45  40926  cdlemk53b  40935  erng1r  40974  tendocnv  41000  dvalveclem  41004  dva0g  41006  dvhgrp  41086  dvhlveclem  41087  dvh0g  41090  cdlemn8  41183  dihordlem7b  41194  dihopelvalcpre  41227  aks6d1c6lem5  42150  mendring  43161  rngccatidALTV  48257  ringccatidALTV  48291