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Theorem arwrcl 15525
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 15508 . . 3  |- Nat  =  ( c  e.  Cat  |->  U.
ran  (Homa
`  c ) )
21dmmptss 5486 . 2  |-  dom Nat  C_  Cat
3 elfvdm 5874 . . 3  |-  ( F  e.  (Nat `  C
)  ->  C  e.  dom Nat )
4 arwrcl.a . . 3  |-  A  =  (Nat `  C )
53, 4eleq2s 2562 . 2  |-  ( F  e.  A  ->  C  e.  dom Nat )
62, 5sseldi 3487 1  |-  ( F  e.  A  ->  C  e.  Cat )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   U.cuni 4235   dom cdm 4988   ran crn 4989   ` cfv 5570   Catccat 15156  Natcarw 15503  Homachoma 15504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fv 5578  df-arw 15508
This theorem is referenced by:  arwhoma  15526  coafval  15545
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