Theorem arwrcl 18011

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 17994	. . 3 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
2	1	dmmptss 6205	. 2 ⊢ dom Arrow ⊆ Cat
3		elfvdm 6874	. . 3 ⊢ (𝐹 ∈ (Arrow‘𝐶) → 𝐶 ∈ dom Arrow)
4		arwrcl.a	. . 3 ⊢ 𝐴 = (Arrow‘𝐶)
5	3, 4	eleq2s 2854	. 2 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ dom Arrow)
6	2, 5	sselid 3919	1 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∪ cuni 4850  dom cdm 5631  ran crn 5632  ‘cfv 6498  Catccat 17630  Arrowcarw 17989  Hom_achoma 17990

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 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fv 6506  df-arw 17994

This theorem is referenced by:  arwhoma  18012  coafval  18031  arweuthinc  50004