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Theorem sscrel 16454
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 16451 . 2 cat = {⟨, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥))}
21relopabi 5234 1 Rel ⊆cat
Colors of variables: wff setvar class
Syntax hints:  wa 384  wex 1702  wcel 1988  wrex 2910  𝒫 cpw 4149   × cxp 5102  Rel wrel 5109   Fn wfn 5871  cfv 5876  Xcixp 7893  cat cssc 16448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-xp 5110  df-rel 5111  df-ssc 16451
This theorem is referenced by:  brssc  16455  ssc1  16462  ssc2  16463  ssctr  16466  issubc  16476
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