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Theorem arwrcl 18002
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 = (Arrowβ€˜πΆ)
Assertion
Ref Expression
arwrcl (𝐹 ∈ 𝐴 β†’ 𝐢 ∈ Cat)

Proof of Theorem arwrcl
StepHypRef Expression
1 df-arw 17985 . . 3 Arrow = (𝑐 ∈ Cat ↦ βˆͺ ran (Homaβ€˜π‘))
21dmmptss 6231 . 2 dom Arrow βŠ† Cat
3 elfvdm 6919 . . 3 (𝐹 ∈ (Arrowβ€˜πΆ) β†’ 𝐢 ∈ dom Arrow)
4 arwrcl.a . . 3 𝐴 = (Arrowβ€˜πΆ)
53, 4eleq2s 2843 . 2 (𝐹 ∈ 𝐴 β†’ 𝐢 ∈ dom Arrow)
62, 5sselid 3973 1 (𝐹 ∈ 𝐴 β†’ 𝐢 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆͺ cuni 4900  dom cdm 5667  ran crn 5668  β€˜cfv 6534  Catccat 17613  Arrowcarw 17980  Homachoma 17981
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fv 6542  df-arw 17985
This theorem is referenced by:  arwhoma  18003  coafval  18022
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