Theorem sscrel 17767

Description: The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Assertion

Ref	Expression
sscrel	⊢ Rel ⊆cat

Proof of Theorem sscrel

Dummy variables ℎ 𝑗 𝑠 𝑡 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ssc 17764	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5820	1 ⊢ Rel ⊆cat

Syntax hints:   ∧ wa 395  ∃wex 1780   ∈ wcel 2105  ∃wrex 3069  𝒫 cpw 4602   × cxp 5674  Rel wrel 5681   Fn wfn 6538  ‘cfv 6543  Xcixp 8897   ⊆_cat cssc 17761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965  df-opab 5211  df-xp 5682  df-rel 5683  df-ssc 17764

This theorem is referenced by:  brssc  17768  ssc1  17775  ssc2  17776  ssctr  17779  issubc  17792