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Theorem dmaf 13881
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 6139 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5453 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 8 . . . . 5  |-  1st  Fn  _V
4 fof 5451 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
51, 4ax-mp 8 . . . . 5  |-  1st : _V
--> _V
6 fnfco 5407 . . . . 5  |-  ( ( 1st  Fn  _V  /\  1st : _V --> _V )  ->  ( 1st  o.  1st )  Fn  _V )
73, 5, 6mp2an 653 . . . 4  |-  ( 1st 
o.  1st )  Fn  _V
8 df-doma 13856 . . . . 5  |- domA 
=  ( 1st  o.  1st )
98fneq1i 5338 . . . 4  |-  (domA  Fn  _V  <->  ( 1st  o. 
1st )  Fn  _V )
107, 9mpbir 200 . . 3  |- domA  Fn  _V
11 ssv 3198 . . 3  |-  A  C_  _V
12 fnssres 5357 . . 3  |-  ( (domA  Fn  _V  /\  A  C_  _V )  ->  (domA  |`  A )  Fn  A
)
1310, 11, 12mp2an 653 . 2  |-  (domA  |`  A )  Fn  A
14 fvres 5542 . . . 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 13879 . . . 4  |-  ( x  e.  A  ->  (domA `  x )  e.  B )
1814, 17eqeltrd 2357 . . 3  |-  ( x  e.  A  ->  (
(domA  |`  A ) `  x
)  e.  B )
1918rgen 2608 . 2  |-  A. x  e.  A  ( (domA  |`  A ) `
 x )  e.  B
20 ffnfv 5685 . 2  |-  ( (domA  |`  A ) : A --> B  <->  ( (domA  |`  A )  Fn  A  /\  A. x  e.  A  (
(domA  |`  A ) `  x
)  e.  B ) )
2113, 19, 20mpbir2an 886 1  |-  (domA  |`  A ) : A --> B
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152    |` cres 4691    o. ccom 4693    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255   1stc1st 6120   Basecbs 13148  domAcdoma 13852  Natcarw 13854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-doma 13856  df-coda 13857  df-homa 13858  df-arw 13859
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