Theorem sscrel 17771

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 17768	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5763	1 ⊢ Rel ⊆cat

Syntax hints:   ∧ wa 396  ∃wex 1786   ∈ wcel 2119  ∃wrex 3063  𝒫 cpw 4529   × cxp 5616  Rel wrel 5623   Fn wfn 6480  ‘cfv 6485  Xcixp 8835   ⊆_cat cssc 17765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-ss 3900  df-opab 5135  df-xp 5624  df-rel 5625  df-ssc 17768

This theorem is referenced by:  brssc  17772  ssc1  17779  ssc2  17780  ssctr  17783  issubc  17793  iinfssc  49547