Theorem sscrel 17688

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 17685	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5774	1 ⊢ Rel ⊆cat

Syntax hints:   ∧ wa 396  ∃wex 1781   ∈ wcel 2106  ∃wrex 3071  𝒫 cpw 4558   × cxp 5629  Rel wrel 5636   Fn wfn 6488  ‘cfv 6493  Xcixp 8831   ⊆_cat cssc 17682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3445  df-in 3915  df-ss 3925  df-opab 5166  df-xp 5637  df-rel 5638  df-ssc 17685

This theorem is referenced by:  brssc  17689  ssc1  17696  ssc2  17697  ssctr  17700  issubc  17713