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Theorem arwrcl 16517
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 16500 . . 3 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
21dmmptss 5548 . 2 dom Arrow ⊆ Cat
3 elfvdm 6130 . . 3 (𝐹 ∈ (Arrow‘𝐶) → 𝐶 ∈ dom Arrow)
4 arwrcl.a . . 3 𝐴 = (Arrow‘𝐶)
53, 4eleq2s 2706 . 2 (𝐹𝐴𝐶 ∈ dom Arrow)
62, 5sseldi 3566 1 (𝐹𝐴𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977   cuni 4372  dom cdm 5038  ran crn 5039  cfv 5804  Catccat 16148  Arrowcarw 16495  Homachoma 16496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-arw 16500
This theorem is referenced by:  arwhoma  16518  coafval  16537
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