MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmaf Structured version   Unicode version

Theorem dmaf 15887
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 6771 . . . . . 6  |-  1st : _V -onto-> _V
2 fofn 5755 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  1st  Fn  _V
4 fof 5753 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
51, 4ax-mp 5 . . . . 5  |-  1st : _V
--> _V
6 fnfco 5708 . . . . 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 15862 . . . . 5  |- domA 
=  ( 1st  o.  1st )
98fneq1i 5631 . . . 4  |-  (domA  Fn  _V  <->  ( 1st  o. 
1st )  Fn  _V )
107, 9mpbir 212 . . 3  |- domA  Fn  _V
11 ssv 3427 . . 3  |-  A  C_  _V
12 fnssres 5650 . . 3  |-  ( (domA  Fn  _V  /\  A  C_  _V )  ->  (domA  |`  A )  Fn  A
)
1310, 11, 12mp2an 676 . 2  |-  (domA  |`  A )  Fn  A
14 fvres 5839 . . . 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 15885 . . . 4  |-  ( x  e.  A  ->  (domA `  x )  e.  B )
1814, 17eqeltrd 2506 . . 3  |-  ( x  e.  A  ->  (
(domA  |`  A ) `  x
)  e.  B )
1918rgen 2724 . 2  |-  A. x  e.  A  ( (domA  |`  A ) `
 x )  e.  B
20 ffnfv 6008 . 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 1872   A.wral 2714   _Vcvv 3022    C_ wss 3379    |` cres 4798    o. ccom 4800    Fn wfn 5539   -->wf 5540   -onto->wfo 5542   ` cfv 5544   1stc1st 6749   Basecbs 15064  domAcdoma 15858  Natcarw 15860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-1st 6751  df-2nd 6752  df-doma 15862  df-coda 15863  df-homa 15864  df-arw 15865
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator