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Theorem sscrel 16858
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 16855 . 2 cat = {⟨, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥))}
21relopabi 5491 1 Rel ⊆cat
Colors of variables: wff setvar class
Syntax hints:  wa 386  wex 1823  wcel 2106  wrex 3090  𝒫 cpw 4378   × cxp 5353  Rel wrel 5360   Fn wfn 6130  cfv 6135  Xcixp 8194  cat cssc 16852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-opab 4949  df-xp 5361  df-rel 5362  df-ssc 16855
This theorem is referenced by:  brssc  16859  ssc1  16866  ssc2  16867  ssctr  16870  issubc  16880
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