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Theorem dfpred2 6330
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 6320 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 dfpred2.1 . . . 4 𝑋 ∈ V
3 iniseg 6114 . . . 4 (𝑋 ∈ V → (𝑅 “ {𝑋}) = {𝑦𝑦𝑅𝑋})
42, 3ax-mp 5 . . 3 (𝑅 “ {𝑋}) = {𝑦𝑦𝑅𝑋}
54ineq2i 4216 . 2 (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
61, 5eqtri 2764 1 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  {cab 2713  Vcvv 3479  cin 3949  {csn 4625   class class class wbr 5142  ccnv 5683  cima 5687  Predcpred 6319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320
This theorem is referenced by:  dfpred3  6331  tz6.26OLD  6368
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