Theorem cdaf 18063

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 8009	. . . . . 6 ⊢ 2nd :V–onto→V
2		fofn 6792	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
4		fo1st 8008	. . . . . 6 ⊢ 1st :V–onto→V
5		fof 6790	. . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1st :V⟶V
7		fnfco 6743	. . . . 5 ⊢ ((2nd Fn V ∧ 1st :V⟶V) → (2nd ∘ 1st ) Fn V)
8	3, 6, 7	mp2an 692	. . . 4 ⊢ (2nd ∘ 1st ) Fn V
9		df-coda 18038	. . . . 5 ⊢ coda = (2nd ∘ 1st )
10	9	fneq1i 6635	. . . 4 ⊢ (coda Fn V ↔ (2nd ∘ 1st ) Fn V)
11	8, 10	mpbir 231	. . 3 ⊢ coda Fn V
12		ssv 3983	. . 3 ⊢ 𝐴 ⊆ V
13		fnssres 6661	. . 3 ⊢ ((coda Fn V ∧ 𝐴 ⊆ V) → (coda ↾ 𝐴) Fn 𝐴)
14	11, 12, 13	mp2an 692	. 2 ⊢ (coda ↾ 𝐴) Fn 𝐴
15		fvres 6895	. . . 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 18061	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (coda‘𝑥) ∈ 𝐵)
19	15, 18	eqeltrd 2834	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵)
20	19	rgen 3053	. 2 ⊢ ∀𝑥 ∈ 𝐴 ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵
21		ffnfv 7109	. 2 ⊢ ((coda ↾ 𝐴):𝐴⟶𝐵 ↔ ((coda ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵))
22	14, 20, 21	mpbir2an 711	1 ⊢ (coda ↾ 𝐴):𝐴⟶𝐵

Syntax hints:   = wceq 1540   ∈ wcel 2108  ∀wral 3051  Vcvv 3459   ⊆ wss 3926   ↾ cres 5656   ∘ ccom 5658   Fn wfn 6526  ⟶wf 6527  –onto→wfo 6529  ‘cfv 6531  1^st c1st 7986  2^nd c2nd 7987  Basecbs 17228  cod_accoda 18034  Arrowcarw 18035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-1st 7988  df-2nd 7989  df-doma 18037  df-coda 18038  df-homa 18039  df-arw 18040

This theorem is referenced by: (None)