Theorem cdaf 16700

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

Hypotheses

Ref	Expression
arwrcl.a	|- A = (Nat ` 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 7189	. . . . . 6 |- 2nd : _V -onto-> _V
2		fofn 6117	. . . . . 6 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3	1, 2	ax-mp 5	. . . . 5 |- 2nd Fn _V
4		fo1st 7188	. . . . . 6 |- 1st : _V -onto-> _V
5		fof 6115	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
6	4, 5	ax-mp 5	. . . . 5 |- 1st : _V --> _V
7		fnfco 6069	. . . . 5 |- ( ( 2nd Fn _V /\ 1st : _V --> _V ) -> ( 2nd o. 1st ) Fn _V )
8	3, 6, 7	mp2an 708	. . . 4 |- ( 2nd o. 1st ) Fn _V
9		df-coda 16675	. . . . 5 |- coda = ( 2nd o. 1st )
10	9	fneq1i 5985	. . . 4 |- (coda Fn _V <-> ( 2nd o. 1st ) Fn _V )
11	8, 10	mpbir 221	. . 3 |- coda Fn _V
12		ssv 3625	. . 3 |- A C_ _V
13		fnssres 6004	. . 3 |- ( (coda Fn _V /\ A C_ _V ) -> (coda |` A ) Fn A )
14	11, 12, 13	mp2an 708	. 2 |- (coda |` A ) Fn A
15		fvres 6207	. . . 4 |- ( x e. A -> ( (coda |` A ) ` x ) = (coda ` x ) )
16		arwrcl.a	. . . . 5 |- A = (Nat ` C )
17		arwdm.b	. . . . 5 |- B = ( Base ` C )
18	16, 17	arwcd 16698	. . . 4 |- ( x e. A -> (coda ` x ) e. B )
19	15, 18	eqeltrd 2701	. . 3 |- ( x e. A -> ( (coda |` A ) ` x ) e. B )
20	19	rgen 2922	. 2 |- A. x e. A ( (coda |` A ) ` x ) e. B
21		ffnfv 6388	. 2 |- ( (coda |` A ) : A --> B <-> ( (coda |` A ) Fn A /\ A. x e. A ( (coda |` A ) ` x ) e. B ) )
22	14, 20, 21	mpbir2an 955	1 |- (coda |` A ) : A --> B

Syntax hints:   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   Basecbs 15857  cod_accoda 16671  Natcarw 16672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-doma 16674  df-coda 16675  df-homa 16676  df-arw 16677

This theorem is referenced by: (None)