Theorem cdaf 17765

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 7852	. . . . . 6 ⊢ 2nd :V–onto→V
2		fofn 6690	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
4		fo1st 7851	. . . . . 6 ⊢ 1st :V–onto→V
5		fof 6688	. . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1st :V⟶V
7		fnfco 6639	. . . . 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 17740	. . . . 5 ⊢ coda = (2nd ∘ 1st )
10	9	fneq1i 6530	. . . 4 ⊢ (coda Fn V ↔ (2nd ∘ 1st ) Fn V)
11	8, 10	mpbir 230	. . 3 ⊢ coda Fn V
12		ssv 3945	. . 3 ⊢ 𝐴 ⊆ V
13		fnssres 6555	. . 3 ⊢ ((coda Fn V ∧ 𝐴 ⊆ V) → (coda ↾ 𝐴) Fn 𝐴)
14	11, 12, 13	mp2an 689	. 2 ⊢ (coda ↾ 𝐴) Fn 𝐴
15		fvres 6793	. . . 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 17763	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (coda‘𝑥) ∈ 𝐵)
19	15, 18	eqeltrd 2839	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵)
20	19	rgen 3074	. 2 ⊢ ∀𝑥 ∈ 𝐴 ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵
21		ffnfv 6992	. 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 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887   ↾ cres 5591   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  –onto→wfo 6431  ‘cfv 6433  1^st c1st 7829  2^nd c2nd 7830  Basecbs 16912  cod_accoda 17736  Arrowcarw 17737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-1st 7831  df-2nd 7832  df-doma 17739  df-coda 17740  df-homa 17741  df-arw 17742

This theorem is referenced by: (None)