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Theorem arwrcl 17416
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 17399 . . 3 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
21dmmptss 6073 . 2 dom Arrow ⊆ Cat
3 elfvdm 6706 . . 3 (𝐹 ∈ (Arrow‘𝐶) → 𝐶 ∈ dom Arrow)
4 arwrcl.a . . 3 𝐴 = (Arrow‘𝐶)
53, 4eleq2s 2851 . 2 (𝐹𝐴𝐶 ∈ dom Arrow)
62, 5sseldi 3875 1 (𝐹𝐴𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114   cuni 4796  dom cdm 5525  ran crn 5526  cfv 6339  Catccat 17038  Arrowcarw 17394  Homachoma 17395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fv 6347  df-arw 17399
This theorem is referenced by:  arwhoma  17417  coafval  17436
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