Theorem sscrel 17905

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 17902	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5801	1 ⊢ Rel ⊆cat

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  ∃wrex 3086  𝒫 cpw 4557   × cxp 5653  Rel wrel 5660   Fn wfn 6528  ‘cfv 6533  Xcixp 8907   ⊆_cat cssc 17899

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 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3916  df-opab 5168  df-xp 5661  df-rel 5662  df-ssc 17902

This theorem is used by:  brssc  17906  ssc1  17913  ssc2  17914  ssctr  17917  issubc  17927  iinfssc  49986