Theorem arwrcl 18234

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 18217	. . 3 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
2	1	dmmptss 6249	. 2 ⊢ dom Arrow ⊆ Cat
3		elfvdm 6930	. . 3 ⊢ (𝐹 ∈ (Arrow‘𝐶) → 𝐶 ∈ dom Arrow)
4		arwrcl.a	. . 3 ⊢ 𝐴 = (Arrow‘𝐶)
5	3, 4	eleq2s 2882	. 2 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ dom Arrow)
6	2, 5	sselid 3911	1 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  ∪ cuni 4870  dom cdm 5655  ran crn 5656  ‘cfv 6547  Catccat 17853  Arrowcarw 18212  Hom_achoma 18213

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fv 6555  df-arw 18217

This theorem is used by:  arwhoma  18235  coafval  18254  arweuthinc  50806