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Theorem arwrcl 14204
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 14187 . . 3  |- Nat  =  ( c  e.  Cat  |->  U.
ran  (Homa
`  c ) )
21dmmptss 5369 . 2  |-  dom Nat  C_  Cat
3 elfvdm 5760 . . 3  |-  ( F  e.  (Nat `  C
)  ->  C  e.  dom Nat )
4 arwrcl.a . . 3  |-  A  =  (Nat `  C )
53, 4eleq2s 2530 . 2  |-  ( F  e.  A  ->  C  e.  dom Nat )
62, 5sseldi 3348 1  |-  ( F  e.  A  ->  C  e.  Cat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   U.cuni 4017   dom cdm 4881   ran crn 4882   ` cfv 5457   Catccat 13894  Natcarw 14182  Homachoma 14183
This theorem is referenced by:  arwhoma  14205  coafval  14224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
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-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fv 5465  df-arw 14187
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