Theorem arwrcl 18127

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 18110	. . 3 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
2	1	dmmptss 6244	. 2 ⊢ dom Arrow ⊆ Cat
3		elfvdm 6919	. . 3 ⊢ (𝐹 ∈ (Arrow‘𝐶) → 𝐶 ∈ dom Arrow)
4		arwrcl.a	. . 3 ⊢ 𝐴 = (Arrow‘𝐶)
5	3, 4	eleq2s 2883	. 2 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ dom Arrow)
6	2, 5	sselid 3936	1 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ∪ cuni 4874  dom cdm 5663  ran crn 5664  ‘cfv 6540  Catccat 17746  Arrowcarw 18105  Hom_achoma 18106

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fv 6548  df-arw 18110

This theorem is used by:  arwhoma  18128  coafval  18147  arweuthinc  50383