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Theorem cdaf 13898
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.
StepHypRef Expression
1 fo2nd 6156 . . . . . 6  |-  2nd : _V -onto-> _V
2 fofn 5469 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 8 . . . . 5  |-  2nd  Fn  _V
4 fo1st 6155 . . . . . 6  |-  1st : _V -onto-> _V
5 fof 5467 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
64, 5ax-mp 8 . . . . 5  |-  1st : _V
--> _V
7 fnfco 5423 . . . . 5  |-  ( ( 2nd  Fn  _V  /\  1st : _V --> _V )  ->  ( 2nd  o.  1st )  Fn  _V )
83, 6, 7mp2an 653 . . . 4  |-  ( 2nd 
o.  1st )  Fn  _V
9 df-coda 13873 . . . . 5  |- coda  =  ( 2nd  o. 
1st )
109fneq1i 5354 . . . 4  |-  (coda  Fn  _V  <->  ( 2nd  o.  1st )  Fn  _V )
118, 10mpbir 200 . . 3  |- coda  Fn  _V
12 ssv 3211 . . 3  |-  A  C_  _V
13 fnssres 5373 . . 3  |-  ( (coda  Fn 
_V  /\  A  C_  _V )  ->  (coda  |`  A )  Fn  A
)
1411, 12, 13mp2an 653 . 2  |-  (coda  |`  A )  Fn  A
15 fvres 5558 . . . 4  |-  ( x  e.  A  ->  (
(coda  |`  A ) `  x
)  =  (coda `  x
) )
16 arwrcl.a . . . . 5  |-  A  =  (Nat `  C )
17 arwdm.b . . . . 5  |-  B  =  ( Base `  C
)
1816, 17arwcd 13896 . . . 4  |-  ( x  e.  A  ->  (coda `  x
)  e.  B )
1915, 18eqeltrd 2370 . . 3  |-  ( x  e.  A  ->  (
(coda  |`  A ) `  x
)  e.  B )
2019rgen 2621 . 2  |-  A. x  e.  A  ( (coda  |`  A ) `
 x )  e.  B
21 ffnfv 5701 . 2  |-  ( (coda  |`  A ) : A --> B 
<->  ( (coda  |`  A )  Fn  A  /\  A. x  e.  A  ( (coda  |`  A ) `  x
)  e.  B ) )
2214, 20, 21mpbir2an 886 1  |-  (coda  |`  A ) : A --> B
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165    |` cres 4707    o. ccom 4709    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271   1stc1st 6136   2ndc2nd 6137   Basecbs 13164  codaccoda 13869  Natcarw 13870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-doma 13872  df-coda 13873  df-homa 13874  df-arw 13875
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