Theorem sscrel 17988

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 17985	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5798	1 ⊢ Rel ⊆cat

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  ∃wrex 3087  𝒫 cpw 4557   × cxp 5649  Rel wrel 5656   Fn wfn 6533  ‘cfv 6538  Xcixp 8925   ⊆_cat cssc 17982

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-ss 3916  df-opab 5168  df-xp 5657  df-rel 5658  df-ssc 17985

This theorem is used by:  brssc  17989  ssc1  17996  ssc2  17997  ssctr  18000  issubc  18010  iinfssc  50164