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Theorem sscrel 16862
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 16859 . 2 cat = {⟨, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥))}
21relopabi 5493 1 Rel ⊆cat
Colors of variables: wff setvar class
Syntax hints:  wa 386  wex 1823  wcel 2107  wrex 3091  𝒫 cpw 4379   × cxp 5355  Rel wrel 5362   Fn wfn 6132  cfv 6137  Xcixp 8196  cat cssc 16856
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 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
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 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-opab 4951  df-xp 5363  df-rel 5364  df-ssc 16859
This theorem is referenced by:  brssc  16863  ssc1  16870  ssc2  16871  ssctr  16874  issubc  16884
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