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Theorem dfpred2 6320
Description: An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 8-Feb-2011.)
Hypothesis
Ref Expression
dfpred2.1 𝑋 ∈ V
Assertion
Ref Expression
dfpred2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
Distinct variable groups:   𝑦,𝑅   𝑦,𝑋
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dfpred2
StepHypRef Expression
1 df-pred 6310 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 dfpred2.1 . . . 4 𝑋 ∈ V
3 iniseg 6106 . . . 4 (𝑋 ∈ V → (𝑅 “ {𝑋}) = {𝑦𝑦𝑅𝑋})
42, 3ax-mp 5 . . 3 (𝑅 “ {𝑋}) = {𝑦𝑦𝑅𝑋}
54ineq2i 4211 . 2 (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
61, 5eqtri 2756 1 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  {cab 2705  Vcvv 3473  cin 3948  {csn 4632   class class class wbr 5152  ccnv 5681  cima 5685  Predcpred 6309
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310
This theorem is referenced by:  dfpred3  6321  tz6.26OLD  6359
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