Theorem cdaf 18015

Description: The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
arwrcl.a	⊢ 𝐴 = (Arrow‘𝐶)
arwdm.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
cdaf	⊢ (coda ↾ 𝐴):𝐴⟶𝐵

Proof of Theorem cdaf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo2nd 7959	. . . . . 6 ⊢ 2nd :V–onto→V
2		fofn 6748	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
4		fo1st 7958	. . . . . 6 ⊢ 1st :V–onto→V
5		fof 6746	. . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1st :V⟶V
7		fnfco 6699	. . . . 5 ⊢ ((2nd Fn V ∧ 1st :V⟶V) → (2nd ∘ 1st ) Fn V)
8	3, 6, 7	mp2an 698	. . . 4 ⊢ (2nd ∘ 1st ) Fn V
9		df-coda 17990	. . . . 5 ⊢ coda = (2nd ∘ 1st )
10	9	fneq1i 6589	. . . 4 ⊢ (coda Fn V ↔ (2nd ∘ 1st ) Fn V)
11	8, 10	mpbir 232	. . 3 ⊢ coda Fn V
12		ssv 3946	. . 3 ⊢ 𝐴 ⊆ V
13		fnssres 6615	. . 3 ⊢ ((coda Fn V ∧ 𝐴 ⊆ V) → (coda ↾ 𝐴) Fn 𝐴)
14	11, 12, 13	mp2an 698	. 2 ⊢ (coda ↾ 𝐴) Fn 𝐴
15		fvres 6853	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((coda ↾ 𝐴)‘𝑥) = (coda‘𝑥))
16		arwrcl.a	. . . . 5 ⊢ 𝐴 = (Arrow‘𝐶)
17		arwdm.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
18	16, 17	arwcd 18013	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (coda‘𝑥) ∈ 𝐵)
19	15, 18	eqeltrd 2840	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵)
20	19	rgen 3056	. 2 ⊢ ∀𝑥 ∈ 𝐴 ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵
21		ffnfv 7067	. 2 ⊢ ((coda ↾ 𝐴):𝐴⟶𝐵 ↔ ((coda ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵))
22	14, 20, 21	mpbir2an 717	1 ⊢ (coda ↾ 𝐴):𝐴⟶𝐵

Syntax hints:   = wceq 1547   ∈ wcel 2119  ∀wral 3054  Vcvv 3432   ⊆ wss 3890   ↾ cres 5627   ∘ ccom 5629   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  1^st c1st 7936  2^nd c2nd 7937  Basecbs 17177  cod_accoda 17986  Arrowcarw 17987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-1st 7938  df-2nd 7939  df-doma 17989  df-coda 17990  df-homa 17991  df-arw 17992

This theorem is referenced by: (None)