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Theorem arwrcl 14162
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 14145 . . 3  |- Nat  =  ( c  e.  Cat  |->  U.
ran  (Homa
`  c ) )
21dmmptss 5333 . 2  |-  dom Nat  C_  Cat
3 elfvdm 5724 . . 3  |-  ( F  e.  (Nat `  C
)  ->  C  e.  dom Nat )
4 arwrcl.a . . 3  |-  A  =  (Nat `  C )
53, 4eleq2s 2504 . 2  |-  ( F  e.  A  ->  C  e.  dom Nat )
62, 5sseldi 3314 1  |-  ( F  e.  A  ->  C  e.  Cat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   U.cuni 3983   dom cdm 4845   ran crn 4846   ` cfv 5421   Catccat 13852  Natcarw 14140  Homachoma 14141
This theorem is referenced by:  arwhoma  14163  coafval  14182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fv 5429  df-arw 14145
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