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Theorem dfpred2 5838
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 5829 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 dfpred2.1 . . . 4 𝑋 ∈ V
3 iniseg 5642 . . . 4 (𝑋 ∈ V → (𝑅 “ {𝑋}) = {𝑦𝑦𝑅𝑋})
42, 3ax-mp 5 . . 3 (𝑅 “ {𝑋}) = {𝑦𝑦𝑅𝑋}
54ineq2i 3942 . 2 (𝐴 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
61, 5eqtri 2770 1 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1620  wcel 2127  {cab 2734  Vcvv 3328  cin 3702  {csn 4309   class class class wbr 4792  ccnv 5253  cima 5257  Predcpred 5828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-xp 5260  df-cnv 5262  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829
This theorem is referenced by:  dfpred3  5839  tz6.26  5860
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