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Theorem dmaf 15895
Description: The domain 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
dmaf  |-  (domA  |`  A ) : A --> B

Proof of Theorem dmaf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fo1st 6827 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5812 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  1st  Fn  _V
4 fof 5810 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
51, 4ax-mp 5 . . . . 5  |-  1st : _V
--> _V
6 fnfco 5765 . . . . 5  |-  ( ( 1st  Fn  _V  /\  1st : _V --> _V )  ->  ( 1st  o.  1st )  Fn  _V )
73, 5, 6mp2an 676 . . . 4  |-  ( 1st 
o.  1st )  Fn  _V
8 df-doma 15870 . . . . 5  |- domA 
=  ( 1st  o.  1st )
98fneq1i 5688 . . . 4  |-  (domA  Fn  _V  <->  ( 1st  o. 
1st )  Fn  _V )
107, 9mpbir 212 . . 3  |- domA  Fn  _V
11 ssv 3490 . . 3  |-  A  C_  _V
12 fnssres 5707 . . 3  |-  ( (domA  Fn  _V  /\  A  C_  _V )  ->  (domA  |`  A )  Fn  A
)
1310, 11, 12mp2an 676 . 2  |-  (domA  |`  A )  Fn  A
14 fvres 5895 . . . 4  |-  ( x  e.  A  ->  (
(domA  |`  A ) `  x
)  =  (domA `  x ) )
15 arwrcl.a . . . . 5  |-  A  =  (Nat `  C )
16 arwdm.b . . . . 5  |-  B  =  ( Base `  C
)
1715, 16arwdm 15893 . . . 4  |-  ( x  e.  A  ->  (domA `  x )  e.  B )
1814, 17eqeltrd 2517 . . 3  |-  ( x  e.  A  ->  (
(domA  |`  A ) `  x
)  e.  B )
1918rgen 2792 . 2  |-  A. x  e.  A  ( (domA  |`  A ) `
 x )  e.  B
20 ffnfv 6064 . 2  |-  ( (domA  |`  A ) : A --> B  <->  ( (domA  |`  A )  Fn  A  /\  A. x  e.  A  (
(domA  |`  A ) `  x
)  e.  B ) )
2113, 19, 20mpbir2an 928 1  |-  (domA  |`  A ) : A --> B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087    C_ wss 3442    |` cres 4856    o. ccom 4858    Fn wfn 5596   -->wf 5597   -onto->wfo 5599   ` cfv 5601   1stc1st 6805   Basecbs 15084  domAcdoma 15866  Natcarw 15868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-1st 6807  df-2nd 6808  df-doma 15870  df-coda 15871  df-homa 15872  df-arw 15873
This theorem is referenced by: (None)
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