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Theorem dfpred2 6154
 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 6145 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 dfpred2.1 . . . 4 𝑋 ∈ V
3 iniseg 5957 . . . 4 (𝑋 ∈ V → (𝑅 “ {𝑋}) = {𝑦𝑦𝑅𝑋})
42, 3ax-mp 5 . . 3 (𝑅 “ {𝑋}) = {𝑦𝑦𝑅𝑋}
54ineq2i 4189 . 2 (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
61, 5eqtri 2848 1 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
 Colors of variables: wff setvar class Syntax hints:   = wceq 1530   ∈ wcel 2107  {cab 2803  Vcvv 3499   ∩ cin 3938  {csn 4563   class class class wbr 5062  ◡ccnv 5552   “ cima 5556  Predcpred 6144 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 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145 This theorem is referenced by:  dfpred3  6155  tz6.26  6176
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