Theorem fcoi1 6078

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 6045	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		df-fn 5891	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3		eqimss 3657	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
4		cnvi 5537	. . . . . . . . . 10 ⊢ ◡ I = I
5	4	reseq1i 5392	. . . . . . . . 9 ⊢ (◡ I ↾ 𝐴) = ( I ↾ 𝐴)
6	5	cnveqi 5297	. . . . . . . 8 ⊢ ◡(◡ I ↾ 𝐴) = ◡( I ↾ 𝐴)
7		cnvresid 5968	. . . . . . . 8 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
8	6, 7	eqtr2i 2645	. . . . . . 7 ⊢ ( I ↾ 𝐴) = ◡(◡ I ↾ 𝐴)
9	8	coeq2i 5282	. . . . . 6 ⊢ (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ ◡(◡ I ↾ 𝐴))
10		cores2 5648	. . . . . 6 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ◡(◡ I ↾ 𝐴)) = (𝐹 ∘ I ))
11	9, 10	syl5eq 2668	. . . . 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 5905	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
14		coi1 5651	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
15	13, 14	syl 17	. . . 4 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
16	12, 15	sylan9eqr 2678	. . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
17	2, 16	sylbi 207	. 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 384   = wceq 1483   ⊆ wss 3574   I cid 5023  ^◡ccnv 5113  dom cdm 5114   ↾ cres 5116   ∘ ccom 5118  Rel wrel 5119  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  fcof1oinvd  6548  mapen  8124  mapfien  8313  hashfacen  13238  cofurid  16551  setccatid  16734  estrccatid  16772  curf2ndf  16887  symgid  17821  f1omvdco2  17868  psgnunilem1  17913  pf1mpf  19716  pf1ind  19719  wilthlem3  24796  hoico1  28615  fmptco1f1o  29434  fcobijfs  29501  reprpmtf1o  30704  ltrncoidN  35414  trlcoabs2N  36010  trlcoat  36011  cdlemg47a  36022  cdlemg46  36023  trljco  36028  tendo1mulr  36059  tendo0co2  36076  cdlemi2  36107  cdlemk2  36120  cdlemk4  36122  cdlemk8  36126  cdlemk53  36245  cdlemk55a  36247  dvhopN  36405  dihopelvalcpre  36537  dihmeetlem1N  36579  dihglblem5apreN  36580  diophrw  37322  mendring  37762  rngccatidALTV  41989  ringccatidALTV  42052