Theorem fcoi1 6698

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

Assertion

Ref	Expression
fcoi1	⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)

Proof of Theorem fcoi1

Step	Hyp	Ref	Expression
1		ffn 6652	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		df-fn 6485	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3		eqimss 3994	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
4		cnvi 6090	. . . . . . . . . 10 ⊢ ◡ I = I
5	4	reseq1i 5926	. . . . . . . . 9 ⊢ (◡ I ↾ 𝐴) = ( I ↾ 𝐴)
6	5	cnveqi 5817	. . . . . . . 8 ⊢ ◡(◡ I ↾ 𝐴) = ◡( I ↾ 𝐴)
7		cnvresid 6561	. . . . . . . 8 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
8	6, 7	eqtr2i 2753	. . . . . . 7 ⊢ ( I ↾ 𝐴) = ◡(◡ I ↾ 𝐴)
9	8	coeq2i 5803	. . . . . 6 ⊢ (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ ◡(◡ I ↾ 𝐴))
10		cores2 6208	. . . . . 6 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ◡(◡ I ↾ 𝐴)) = (𝐹 ∘ I ))
11	9, 10	eqtrid 2776	. . . . 5 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
12	3, 11	syl 17	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
13		funrel 6499	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
14		coi1 6211	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
15	13, 14	syl 17	. . . 4 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
16	12, 15	sylan9eqr 2786	. . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
17	2, 16	sylbi 217	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
18	1, 17	syl 17	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ⊆ wss 3903   I cid 5513  ^◡ccnv 5618  dom cdm 5619   ↾ cres 5621   ∘ ccom 5623  Rel wrel 5624  Fun wfun 6476   Fn wfn 6477  ⟶wf 6478

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-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-fun 6484  df-fn 6485  df-f 6486

This theorem is referenced by:  fcof1oinvd  7230  mapen  9058  mapfien  9298  hashfacen  14361  cofurid  17798  setccatid  17991  estrccatid  18038  curf2ndf  18153  efmndid  18762  efmndmnd  18763  f1omvdco2  19327  psgnunilem1  19372  pf1mpf  22237  pf1ind  22240  wilthlem3  26978  hoico1  31700  fmptco1f1o  32576  fcobijfs  32665  cycpmconjslem2  33097  cycpmconjs  33098  cyc3conja  33099  1arithidomlem2  33473  mplvrpmga  33546  reprpmtf1o  34594  ltrncoidN  40107  trlcoabs2N  40701  trlcoat  40702  cdlemg47a  40713  cdlemg46  40714  trljco  40719  tendo1mulr  40750  tendo0co2  40767  cdlemi2  40798  cdlemk2  40811  cdlemk4  40813  cdlemk8  40817  cdlemk53  40936  cdlemk55a  40938  dvhopN  41095  dihopelvalcpre  41227  dihmeetlem1N  41269  dihglblem5apreN  41270  diophrw  42732  mendring  43161  rngccatidALTV  48256  ringccatidALTV  48290