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Theorem predep 6322
Description: The predecessor under the membership relation is equivalent to an intersection. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
predep (𝑋𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))

Proof of Theorem predep
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-pred 6291 . 2 Pred( E , 𝐴, 𝑋) = (𝐴 ∩ ( E “ {𝑋}))
2 relcnv 6094 . . . . 5 Rel E
3 relimasn 6074 . . . . 5 (Rel E → ( E “ {𝑋}) = {𝑦𝑋 E 𝑦})
42, 3ax-mp 5 . . . 4 ( E “ {𝑋}) = {𝑦𝑋 E 𝑦}
5 brcnvg 5870 . . . . . . 7 ((𝑋𝐵𝑦 ∈ V) → (𝑋 E 𝑦𝑦 E 𝑋))
65elvd 3473 . . . . . 6 (𝑋𝐵 → (𝑋 E 𝑦𝑦 E 𝑋))
7 epelg 5572 . . . . . 6 (𝑋𝐵 → (𝑦 E 𝑋𝑦𝑋))
86, 7bitrd 279 . . . . 5 (𝑋𝐵 → (𝑋 E 𝑦𝑦𝑋))
98eqabcdv 2860 . . . 4 (𝑋𝐵 → {𝑦𝑋 E 𝑦} = 𝑋)
104, 9eqtrid 2776 . . 3 (𝑋𝐵 → ( E “ {𝑋}) = 𝑋)
1110ineq2d 4205 . 2 (𝑋𝐵 → (𝐴 ∩ ( E “ {𝑋})) = (𝐴𝑋))
121, 11eqtrid 2776 1 (𝑋𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  {cab 2701  Vcvv 3466  cin 3940  {csn 4621   class class class wbr 5139   E cep 5570  ccnv 5666  cima 5670  Rel wrel 5672  Predcpred 6290
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-eprel 5571  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291
This theorem is referenced by:  trpred  6323  predonOLD  7768  omsindsOLD  7871
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