Theorem sscrel 17822

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 17819	. 2 ⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
2	1	relopabiv 5817	1 ⊢ Rel ⊆cat

Syntax hints:   ∧ wa 394  ∃wex 1774   ∈ wcel 2099  ∃wrex 3060  𝒫 cpw 4598   × cxp 5671  Rel wrel 5678   Fn wfn 6539  ‘cfv 6544  Xcixp 8916   ⊆_cat cssc 17816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3465  df-ss 3964  df-opab 5207  df-xp 5679  df-rel 5680  df-ssc 17819

This theorem is referenced by:  brssc  17823  ssc1  17830  ssc2  17831  ssctr  17834  issubc  17847