Theorem sscrel 17870

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 17867	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5837	1 ⊢ Rel ⊆cat

Syntax hints:   ∧ wa 395  ∃wex 1778   ∈ wcel 2108  ∃wrex 3070  𝒫 cpw 4608   × cxp 5691  Rel wrel 5698   Fn wfn 6564  ‘cfv 6569  Xcixp 8945   ⊆_cat cssc 17864

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3483  df-ss 3983  df-opab 5214  df-xp 5699  df-rel 5700  df-ssc 17867

This theorem is referenced by:  brssc  17871  ssc1  17878  ssc2  17879  ssctr  17882  issubc  17895