Theorem fcoi1 5547

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 5508	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		df-fn 5355	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3		eqimss 3292	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
4		cnvi 5167	. . . . . . . . . 10 ⊢ ◡ I = I
5	4	reseq1i 5034	. . . . . . . . 9 ⊢ (◡ I ↾ 𝐴) = ( I ↾ 𝐴)
6	5	cnveqi 4930	. . . . . . . 8 ⊢ ◡(◡ I ↾ 𝐴) = ◡( I ↾ 𝐴)
7		cnvresid 5430	. . . . . . . 8 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
8	6, 7	eqtr2i 2254	. . . . . . 7 ⊢ ( I ↾ 𝐴) = ◡(◡ I ↾ 𝐴)
9	8	coeq2i 4915	. . . . . 6 ⊢ (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ ◡(◡ I ↾ 𝐴))
10		cores2 5275	. . . . . 6 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ◡(◡ I ↾ 𝐴)) = (𝐹 ∘ I ))
11	9, 10	eqtrid 2277	. . . . 5 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
12	3, 11	syl 14	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I ))
13		funrel 5369	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
14		coi1 5278	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
15	13, 14	syl 14	. . . 4 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
16	12, 15	sylan9eqr 2287	. . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
17	2, 16	sylbi 121	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
18	1, 17	syl 14	1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ⊆ wss 3211   I cid 4409  ^◡ccnv 4748  dom cdm 4749   ↾ cres 4751   ∘ ccom 4753  Rel wrel 4754  Fun wfun 5346   Fn wfn 5347  ⟶wf 5348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-fun 5354  df-fn 5355  df-f 5356

This theorem is referenced by:  fcof1o  5962  mapen  7099  hashfacen  11208  gsumgfsum1  16863