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Theorem predss 5848
Description: The predecessor class of 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predss Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴

Proof of Theorem predss
StepHypRef Expression
1 df-pred 5841 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 inss1 3976 . 2 (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐴
31, 2eqsstri 3776 1 Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  cin 3714  wss 3715  {csn 4321  ccnv 5265  cima 5269  Predcpred 5840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-pred 5841
This theorem is referenced by:  wfr3g  7582  wfrlem4  7587  wfrlem10  7593  trpredlem1  32032  wsuclem  32076  frr3g  32085  frrlem4  32089
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