Theorem sscrel 17874

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 17871	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5844	1 ⊢ Rel ⊆cat

Syntax hints:   ∧ wa 395  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  𝒫 cpw 4622   × cxp 5698  Rel wrel 5705   Fn wfn 6568  ‘cfv 6573  Xcixp 8955   ⊆_cat cssc 17868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-ss 3993  df-opab 5229  df-xp 5706  df-rel 5707  df-ssc 17871

This theorem is referenced by:  brssc  17875  ssc1  17882  ssc2  17883  ssctr  17886  issubc  17899