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Theorem dmaf 14206
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 6368 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5657 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 8 . . . . 5  |-  1st  Fn  _V
4 fof 5655 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
51, 4ax-mp 8 . . . . 5  |-  1st : _V
--> _V
6 fnfco 5611 . . . . 5  |-  ( ( 1st  Fn  _V  /\  1st : _V --> _V )  ->  ( 1st  o.  1st )  Fn  _V )
73, 5, 6mp2an 655 . . . 4  |-  ( 1st 
o.  1st )  Fn  _V
8 df-doma 14181 . . . . 5  |- domA 
=  ( 1st  o.  1st )
98fneq1i 5541 . . . 4  |-  (domA  Fn  _V  <->  ( 1st  o. 
1st )  Fn  _V )
107, 9mpbir 202 . . 3  |- domA  Fn  _V
11 ssv 3370 . . 3  |-  A  C_  _V
12 fnssres 5560 . . 3  |-  ( (domA  Fn  _V  /\  A  C_  _V )  ->  (domA  |`  A )  Fn  A
)
1310, 11, 12mp2an 655 . 2  |-  (domA  |`  A )  Fn  A
14 fvres 5747 . . . 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 14204 . . . 4  |-  ( x  e.  A  ->  (domA `  x )  e.  B )
1814, 17eqeltrd 2512 . . 3  |-  ( x  e.  A  ->  (
(domA  |`  A ) `  x
)  e.  B )
1918rgen 2773 . 2  |-  A. x  e.  A  ( (domA  |`  A ) `
 x )  e.  B
20 ffnfv 5896 . 2  |-  ( (domA  |`  A ) : A --> B  <->  ( (domA  |`  A )  Fn  A  /\  A. x  e.  A  (
(domA  |`  A ) `  x
)  e.  B ) )
2113, 19, 20mpbir2an 888 1  |-  (domA  |`  A ) : A --> B
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322    |` cres 4882    o. ccom 4884    Fn wfn 5451   -->wf 5452   -onto->wfo 5454   ` cfv 5456   1stc1st 6349   Basecbs 13471  domAcdoma 14177  Natcarw 14179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-1st 6351  df-2nd 6352  df-doma 14181  df-coda 14182  df-homa 14183  df-arw 14184
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