Theorem sscrel 17828

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 17825	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5810	1 ⊢ Rel ⊆cat

Syntax hints:   ∧ wa 395  ∃wex 1778   ∈ wcel 2107  ∃wrex 3059  𝒫 cpw 4580   × cxp 5663  Rel wrel 5670   Fn wfn 6536  ‘cfv 6541  Xcixp 8919   ⊆_cat cssc 17822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-ss 3948  df-opab 5186  df-xp 5671  df-rel 5672  df-ssc 17825

This theorem is referenced by:  brssc  17829  ssc1  17836  ssc2  17837  ssctr  17840  issubc  17851