Theorem cdaf 18007

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 8000	. . . . . 6 ⊢ 2nd :V–onto→V
2		fofn 6807	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
4		fo1st 7999	. . . . . 6 ⊢ 1st :V–onto→V
5		fof 6805	. . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1st :V⟶V
7		fnfco 6756	. . . . 5 ⊢ ((2nd Fn V ∧ 1st :V⟶V) → (2nd ∘ 1st ) Fn V)
8	3, 6, 7	mp2an 689	. . . 4 ⊢ (2nd ∘ 1st ) Fn V
9		df-coda 17982	. . . . 5 ⊢ coda = (2nd ∘ 1st )
10	9	fneq1i 6646	. . . 4 ⊢ (coda Fn V ↔ (2nd ∘ 1st ) Fn V)
11	8, 10	mpbir 230	. . 3 ⊢ coda Fn V
12		ssv 4006	. . 3 ⊢ 𝐴 ⊆ V
13		fnssres 6673	. . 3 ⊢ ((coda Fn V ∧ 𝐴 ⊆ V) → (coda ↾ 𝐴) Fn 𝐴)
14	11, 12, 13	mp2an 689	. 2 ⊢ (coda ↾ 𝐴) Fn 𝐴
15		fvres 6910	. . . 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 18005	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (coda‘𝑥) ∈ 𝐵)
19	15, 18	eqeltrd 2832	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵)
20	19	rgen 3062	. 2 ⊢ ∀𝑥 ∈ 𝐴 ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵
21		ffnfv 7120	. 2 ⊢ ((coda ↾ 𝐴):𝐴⟶𝐵 ↔ ((coda ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵))
22	14, 20, 21	mpbir2an 708	1 ⊢ (coda ↾ 𝐴):𝐴⟶𝐵

Syntax hints:   = wceq 1540   ∈ wcel 2105  ∀wral 3060  Vcvv 3473   ⊆ wss 3948   ↾ cres 5678   ∘ ccom 5680   Fn wfn 6538  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543  1^st c1st 7977  2^nd c2nd 7978  Basecbs 17151  cod_accoda 17978  Arrowcarw 17979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-1st 7979  df-2nd 7980  df-doma 17981  df-coda 17982  df-homa 17983  df-arw 17984

This theorem is referenced by: (None)