Theorem arwrcl 17980

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 17963	. . 3 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
2	1	dmmptss 6207	. 2 ⊢ dom Arrow ⊆ Cat
3		elfvdm 6876	. . 3 ⊢ (𝐹 ∈ (Arrow‘𝐶) → 𝐶 ∈ dom Arrow)
4		arwrcl.a	. . 3 ⊢ 𝐴 = (Arrow‘𝐶)
5	3, 4	eleq2s 2855	. 2 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ dom Arrow)
6	2, 5	sselid 3933	1 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∪ cuni 4865  dom cdm 5632  ran crn 5633  ‘cfv 6500  Catccat 17599  Arrowcarw 17958  Hom_achoma 17959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fv 6508  df-arw 17963

This theorem is referenced by:  arwhoma  17981  coafval  18000  arweuthinc  49892