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Theorem arwrcl 18033
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 18016 . . 3 Arrow = (𝑐 ∈ Cat ↦ βˆͺ ran (Homaβ€˜π‘))
21dmmptss 6245 . 2 dom Arrow βŠ† Cat
3 elfvdm 6934 . . 3 (𝐹 ∈ (Arrowβ€˜πΆ) β†’ 𝐢 ∈ dom Arrow)
4 arwrcl.a . . 3 𝐴 = (Arrowβ€˜πΆ)
53, 4eleq2s 2847 . 2 (𝐹 ∈ 𝐴 β†’ 𝐢 ∈ dom Arrow)
62, 5sselid 3978 1 (𝐹 ∈ 𝐴 β†’ 𝐢 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  βˆͺ cuni 4908  dom cdm 5678  ran crn 5679  β€˜cfv 6548  Catccat 17644  Arrowcarw 18011  Homachoma 18012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fv 6556  df-arw 18016
This theorem is referenced by:  arwhoma  18034  coafval  18053
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