Theorem sscrel 17781

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 17778	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5785	1 ⊢ Rel ⊆cat

Syntax hints:   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  ∃wrex 3054  𝒫 cpw 4565   × cxp 5638  Rel wrel 5645   Fn wfn 6508  ‘cfv 6513  Xcixp 8872   ⊆_cat cssc 17775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-ss 3933  df-opab 5172  df-xp 5646  df-rel 5647  df-ssc 17778

This theorem is referenced by:  brssc  17782  ssc1  17789  ssc2  17790  ssctr  17793  issubc  17803  iinfssc  49034