Theorem fcoi2 5262

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 5085	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		cores 5000	. . 3 ⊢ (ran 𝐹 ⊆ 𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = ( I ∘ 𝐹))
3		fnrel 5179	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
4		coi2 5013	. . . 4 ⊢ (Rel 𝐹 → ( I ∘ 𝐹) = 𝐹)
5	3, 4	syl 14	. . 3 ⊢ (𝐹 Fn 𝐴 → ( I ∘ 𝐹) = 𝐹)
6	2, 5	sylan9eqr 2169	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
7	1, 6	sylbi 120	1 ⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1314   ⊆ wss 3037   I cid 4170  ran crn 4500   ↾ cres 4501   ∘ ccom 4503  Rel wrel 4504   Fn wfn 5076  ⟶wf 5077

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-fun 5083  df-fn 5084  df-f 5085

This theorem is referenced by:  fcof1o  5644  mapen  6693  hashfacen  10472