Theorem cdaf 17013

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 7423	. . . . . 6 ⊢ 2nd :V–onto→V
2		fofn 6334	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
4		fo1st 7422	. . . . . 6 ⊢ 1st :V–onto→V
5		fof 6332	. . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1st :V⟶V
7		fnfco 6285	. . . . 5 ⊢ ((2nd Fn V ∧ 1st :V⟶V) → (2nd ∘ 1st ) Fn V)
8	3, 6, 7	mp2an 684	. . . 4 ⊢ (2nd ∘ 1st ) Fn V
9		df-coda 16988	. . . . 5 ⊢ coda = (2nd ∘ 1st )
10	9	fneq1i 6197	. . . 4 ⊢ (coda Fn V ↔ (2nd ∘ 1st ) Fn V)
11	8, 10	mpbir 223	. . 3 ⊢ coda Fn V
12		ssv 3822	. . 3 ⊢ 𝐴 ⊆ V
13		fnssres 6216	. . 3 ⊢ ((coda Fn V ∧ 𝐴 ⊆ V) → (coda ↾ 𝐴) Fn 𝐴)
14	11, 12, 13	mp2an 684	. 2 ⊢ (coda ↾ 𝐴) Fn 𝐴
15		fvres 6431	. . . 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 17011	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (coda‘𝑥) ∈ 𝐵)
19	15, 18	eqeltrd 2879	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵)
20	19	rgen 3104	. 2 ⊢ ∀𝑥 ∈ 𝐴 ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵
21		ffnfv 6615	. 2 ⊢ ((coda ↾ 𝐴):𝐴⟶𝐵 ↔ ((coda ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵))
22	14, 20, 21	mpbir2an 703	1 ⊢ (coda ↾ 𝐴):𝐴⟶𝐵

Syntax hints:   = wceq 1653   ∈ wcel 2157  ∀wral 3090  Vcvv 3386   ⊆ wss 3770   ↾ cres 5315   ∘ ccom 5317   Fn wfn 6097  ⟶wf 6098  –onto→wfo 6100  ‘cfv 6102  1^st c1st 7400  2^nd c2nd 7401  Basecbs 16183  cod_accoda 16984  Arrowcarw 16985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-ov 6882  df-1st 7402  df-2nd 7403  df-doma 16987  df-coda 16988  df-homa 16989  df-arw 16990

This theorem is referenced by: (None)