Theorem arwrcl 17994

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	⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat)

Proof of Theorem arwrcl

Step	Hyp	Ref	Expression
1		df-arw 17977	. . 3 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
2	1	dmmptss 6241	. 2 ⊢ dom Arrow ⊆ Cat
3		elfvdm 6929	. . 3 ⊢ (𝐹 ∈ (Arrow‘𝐶) → 𝐶 ∈ dom Arrow)
4		arwrcl.a	. . 3 ⊢ 𝐴 = (Arrow‘𝐶)
5	3, 4	eleq2s 2852	. 2 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ dom Arrow)
6	2, 5	sselid 3981	1 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  ∪ cuni 4909  dom cdm 5677  ran crn 5678  ‘cfv 6544  Catccat 17608  Arrowcarw 17972  Hom_achoma 17973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-arw 17977

This theorem is referenced by:  arwhoma  17995  coafval  18014