Theorem sscrel 17735

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 17732	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5767	1 ⊢ Rel ⊆cat

Syntax hints:   ∧ wa 395  ∃wex 1780   ∈ wcel 2113  ∃wrex 3058  𝒫 cpw 4552   × cxp 5620  Rel wrel 5627   Fn wfn 6485  ‘cfv 6490  Xcixp 8833   ⊆_cat cssc 17729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-ss 3916  df-opab 5159  df-xp 5628  df-rel 5629  df-ssc 17732

This theorem is referenced by:  brssc  17736  ssc1  17743  ssc2  17744  ssctr  17747  issubc  17757  iinfssc  49244