Theorem fcoi1 6723

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 6676	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		df-fn 6509	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3		eqimss 3985	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
4		cnvi 5846	. . . . . . . . . 10 ⊢ ◡ I = I
5	4	reseq1i 5950	. . . . . . . . 9 ⊢ (◡ I ↾ 𝐴) = ( I ↾ 𝐴)
6	5	cnveqi 5835	. . . . . . . 8 ⊢ ◡(◡ I ↾ 𝐴) = ◡( I ↾ 𝐴)
7		cnvresid 6585	. . . . . . . 8 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
8	6, 7	eqtr2i 2776	. . . . . . 7 ⊢ ( I ↾ 𝐴) = ◡(◡ I ↾ 𝐴)
9	8	coeq2i 5821	. . . . . 6 ⊢ (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ ◡(◡ I ↾ 𝐴))
10		cores2 6232	. . . . . 6 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ◡(◡ I ↾ 𝐴)) = (𝐹 ∘ I ))
11	9, 10	eqtrid 2799	. . . . 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 6523	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
14		coi1 6235	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
15	13, 14	syl 17	. . . 4 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
16	12, 15	sylan9eqr 2809	. . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
17	2, 16	sylbi 219	. 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 398   = wceq 1550   ⊆ wss 3895   I cid 5530  ^◡ccnv 5635  dom cdm 5636   ↾ cres 5638   ∘ ccom 5640  Rel wrel 5641  Fun wfun 6500   Fn wfn 6501  ⟶wf 6502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-br 5091  df-opab 5153  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-fun 6508  df-fn 6509  df-f 6510

This theorem is referenced by:  fcof1oinvd  7262  mapen  9098  mapfien  9340  hashfacen  14453  cofurid  17896  setccatid  18089  estrccatid  18136  curf2ndf  18251  efmndid  18894  efmndmnd  18895  f1omvdco2  19460  psgnunilem1  19505  pf1mpf  22384  pf1ind  22387  wilthlem3  27100  hoico1  31894  fmptco1f1o  32774  fcobijfs  32862  cocnvf1o  32870  cycpmconjslem2  33285  cycpmconjs  33286  cyc3conja  33287  1arithidomlem2  33676  mplvrpmga  33786  mplvrpmrhm  33788  reprpmtf1o  34867  ltrncoidN  40690  trlcoabs2N  41284  trlcoat  41285  cdlemg47a  41296  cdlemg46  41297  trljco  41302  tendo1mulr  41333  tendo0co2  41350  cdlemi2  41381  cdlemk2  41394  cdlemk4  41396  cdlemk8  41400  cdlemk53  41519  cdlemk55a  41521  dvhopN  41678  dihopelvalcpre  41810  dihmeetlem1N  41852  dihglblem5apreN  41853  diophrw  43278  mendring  43703  rngccatidALTV  48832  ringccatidALTV  48866