Theorem sscrel 18000

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 17997	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5802	1 ⊢ Rel ⊆cat

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1811   ∈ wcel 2146  ∃wrex 3089  𝒫 cpw 4559   × cxp 5652  Rel wrel 5659   Fn wfn 6541  ‘cfv 6546  Xcixp 8936   ⊆_cat cssc 17994

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3455  df-ss 3900  df-opab 5170  df-xp 5660  df-rel 5661  df-ssc 17997

This theorem is used by:  brssc  18001  ssc1  18008  ssc2  18009  ssctr  18012  issubc  18022  iinfssc  50310