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Theorem arwrcl 15014
Description: The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
arwrcl.a  |-  A  =  (Nat `  C )
Assertion
Ref Expression
arwrcl  |-  ( F  e.  A  ->  C  e.  Cat )

Proof of Theorem arwrcl
StepHypRef Expression
1 df-arw 14997 . . 3  |- Nat  =  ( c  e.  Cat  |->  U.
ran  (Homa
`  c ) )
21dmmptss 5432 . 2  |-  dom Nat  C_  Cat
3 elfvdm 5815 . . 3  |-  ( F  e.  (Nat `  C
)  ->  C  e.  dom Nat )
4 arwrcl.a . . 3  |-  A  =  (Nat `  C )
53, 4eleq2s 2559 . 2  |-  ( F  e.  A  ->  C  e.  dom Nat )
62, 5sseldi 3452 1  |-  ( F  e.  A  ->  C  e.  Cat )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   U.cuni 4189   dom cdm 4938   ran crn 4939   ` cfv 5516   Catccat 14704  Natcarw 14992  Homachoma 14993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-xp 4944  df-rel 4945  df-cnv 4946  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fv 5524  df-arw 14997
This theorem is referenced by:  arwhoma  15015  coafval  15034
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