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Theorem arwrcl 16895
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 16878 . . 3 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
21dmmptss 5792 . 2 dom Arrow ⊆ Cat
3 elfvdm 6381 . . 3 (𝐹 ∈ (Arrow‘𝐶) → 𝐶 ∈ dom Arrow)
4 arwrcl.a . . 3 𝐴 = (Arrow‘𝐶)
53, 4eleq2s 2857 . 2 (𝐹𝐴𝐶 ∈ dom Arrow)
62, 5sseldi 3742 1 (𝐹𝐴𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139   cuni 4588  dom cdm 5266  ran crn 5267  cfv 6049  Catccat 16526  Arrowcarw 16873  Homachoma 16874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fv 6057  df-arw 16878
This theorem is referenced by:  arwhoma  16896  coafval  16915
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