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Theorem sscrel 17078
 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 𝑗 𝑠 𝑡 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ssc 17075 . 2 cat = {⟨, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥))}
21relopabi 5662 1 Rel ⊆cat
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399  ∃wex 1781   ∈ wcel 2112  ∃wrex 3110  𝒫 cpw 4500   × cxp 5521  Rel wrel 5528   Fn wfn 6323  ‘cfv 6328  Xcixp 8448   ⊆cat cssc 17072 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5096  df-xp 5529  df-rel 5530  df-ssc 17075 This theorem is referenced by:  brssc  17079  ssc1  17086  ssc2  17087  ssctr  17090  issubc  17100
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