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Theorem dmaf 15233
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 6804 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5796 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  1st  Fn  _V
4 fof 5794 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
51, 4ax-mp 5 . . . . 5  |-  1st : _V
--> _V
6 fnfco 5749 . . . . 5  |-  ( ( 1st  Fn  _V  /\  1st : _V --> _V )  ->  ( 1st  o.  1st )  Fn  _V )
73, 5, 6mp2an 672 . . . 4  |-  ( 1st 
o.  1st )  Fn  _V
8 df-doma 15208 . . . . 5  |- domA 
=  ( 1st  o.  1st )
98fneq1i 5674 . . . 4  |-  (domA  Fn  _V  <->  ( 1st  o. 
1st )  Fn  _V )
107, 9mpbir 209 . . 3  |- domA  Fn  _V
11 ssv 3524 . . 3  |-  A  C_  _V
12 fnssres 5693 . . 3  |-  ( (domA  Fn  _V  /\  A  C_  _V )  ->  (domA  |`  A )  Fn  A
)
1310, 11, 12mp2an 672 . 2  |-  (domA  |`  A )  Fn  A
14 fvres 5879 . . . 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 15231 . . . 4  |-  ( x  e.  A  ->  (domA `  x )  e.  B )
1814, 17eqeltrd 2555 . . 3  |-  ( x  e.  A  ->  (
(domA  |`  A ) `  x
)  e.  B )
1918rgen 2824 . 2  |-  A. x  e.  A  ( (domA  |`  A ) `
 x )  e.  B
20 ffnfv 6046 . 2  |-  ( (domA  |`  A ) : A --> B  <->  ( (domA  |`  A )  Fn  A  /\  A. x  e.  A  (
(domA  |`  A ) `  x
)  e.  B ) )
2113, 19, 20mpbir2an 918 1  |-  (domA  |`  A ) : A --> B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476    |` cres 5001    o. ccom 5003    Fn wfn 5582   -->wf 5583   -onto->wfo 5585   ` cfv 5587   1stc1st 6782   Basecbs 14489  domAcdoma 15204  Natcarw 15206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-1st 6784  df-2nd 6785  df-doma 15208  df-coda 15209  df-homa 15210  df-arw 15211
This theorem is referenced by: (None)
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