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Theorem predss 6148
 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 6141 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 inss1 4203 . 2 (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐴
31, 2eqsstri 3999 1 Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∩ cin 3933   ⊆ wss 3934  {csn 4559  ◡ccnv 5547   “ cima 5551  Predcpred 6140 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-in 3941  df-ss 3950  df-pred 6141 This theorem is referenced by:  wfr3g  7945  wfrlem4  7950  wfrlem10  7956  trpredlem1  33054  wsuclem  33100  frr3g  33109  fpr3g  33110  frrlem4  33114  frrlem13  33123  fpr1  33127
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