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Theorem predep 6351
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 6321 . 2 Pred( E , 𝐴, 𝑋) = (𝐴 ∩ ( E “ {𝑋}))
2 relcnv 6122 . . . . 5 Rel E
3 relimasn 6103 . . . . 5 (Rel E → ( E “ {𝑋}) = {𝑦𝑋 E 𝑦})
42, 3ax-mp 5 . . . 4 ( E “ {𝑋}) = {𝑦𝑋 E 𝑦}
5 brcnvg 5890 . . . . . . 7 ((𝑋𝐵𝑦 ∈ V) → (𝑋 E 𝑦𝑦 E 𝑋))
65elvd 3486 . . . . . 6 (𝑋𝐵 → (𝑋 E 𝑦𝑦 E 𝑋))
7 epelg 5585 . . . . . 6 (𝑋𝐵 → (𝑦 E 𝑋𝑦𝑋))
86, 7bitrd 279 . . . . 5 (𝑋𝐵 → (𝑋 E 𝑦𝑦𝑋))
98eqabcdv 2876 . . . 4 (𝑋𝐵 → {𝑦𝑋 E 𝑦} = 𝑋)
104, 9eqtrid 2789 . . 3 (𝑋𝐵 → ( E “ {𝑋}) = 𝑋)
1110ineq2d 4220 . 2 (𝑋𝐵 → (𝐴 ∩ ( E “ {𝑋})) = (𝐴𝑋))
121, 11eqtrid 2789 1 (𝑋𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  cin 3950  {csn 4626   class class class wbr 5143   E cep 5583  ccnv 5684  cima 5688  Rel wrel 5690  Predcpred 6320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321
This theorem is referenced by:  trpred  6352
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