Theorem fcoi1 5552

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 5513	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		df-fn 5360	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3		eqimss 3296	. . . . 5 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
4		cnvi 5172	. . . . . . . . . 10 ⊢ ◡ I = I
5	4	reseq1i 5039	. . . . . . . . 9 ⊢ (◡ I ↾ 𝐴) = ( I ↾ 𝐴)
6	5	cnveqi 4935	. . . . . . . 8 ⊢ ◡(◡ I ↾ 𝐴) = ◡( I ↾ 𝐴)
7		cnvresid 5435	. . . . . . . 8 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
8	6, 7	eqtr2i 2256	. . . . . . 7 ⊢ ( I ↾ 𝐴) = ◡(◡ I ↾ 𝐴)
9	8	coeq2i 4920	. . . . . 6 ⊢ (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ ◡(◡ I ↾ 𝐴))
10		cores2 5280	. . . . . 6 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ◡(◡ I ↾ 𝐴)) = (𝐹 ∘ I ))
11	9, 10	eqtrid 2279	. . . . 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 5374	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
14		coi1 5283	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
15	13, 14	syl 14	. . . 4 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
16	12, 15	sylan9eqr 2289	. . 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 3214   I cid 4414  ^◡ccnv 4753  dom cdm 4754   ↾ cres 4756   ∘ ccom 4758  Rel wrel 4759  Fun wfun 5351   Fn wfn 5352  ⟶wf 5353

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-fun 5359  df-fn 5360  df-f 5361

This theorem is referenced by:  fcof1o  5968  mapen  7112  hashfacen  11233  gsumgfsum1  14103