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Theorem arwrcl 17296
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 = (Arrow‘𝐶)
Assertion
Ref Expression
arwrcl (𝐹𝐴𝐶 ∈ Cat)

Proof of Theorem arwrcl
StepHypRef Expression
1 df-arw 17279 . . 3 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
21dmmptss 6092 . 2 dom Arrow ⊆ Cat
3 elfvdm 6698 . . 3 (𝐹 ∈ (Arrow‘𝐶) → 𝐶 ∈ dom Arrow)
4 arwrcl.a . . 3 𝐴 = (Arrow‘𝐶)
53, 4eleq2s 2935 . 2 (𝐹𝐴𝐶 ∈ dom Arrow)
62, 5sseldi 3968 1 (𝐹𝐴𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107   cuni 4836  dom cdm 5553  ran crn 5554  cfv 6351  Catccat 16927  Arrowcarw 17274  Homachoma 17275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-xp 5559  df-rel 5560  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fv 6359  df-arw 17279
This theorem is referenced by:  arwhoma  17297  coafval  17316
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