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Theorem predep 5623
 Description: The predecessor under the epsilon relationship 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 5597 . 2 Pred( E , 𝐴, 𝑋) = (𝐴 ∩ ( E “ {𝑋}))
2 relcnv 5422 . . . . 5 Rel E
3 relimasn 5407 . . . . 5 (Rel E → ( E “ {𝑋}) = {𝑦𝑋 E 𝑦})
42, 3ax-mp 5 . . . 4 ( E “ {𝑋}) = {𝑦𝑋 E 𝑦}
5 vex 3176 . . . . . . 7 𝑦 ∈ V
6 brcnvg 5225 . . . . . . 7 ((𝑋𝐵𝑦 ∈ V) → (𝑋 E 𝑦𝑦 E 𝑋))
75, 6mpan2 703 . . . . . 6 (𝑋𝐵 → (𝑋 E 𝑦𝑦 E 𝑋))
8 epelg 4950 . . . . . 6 (𝑋𝐵 → (𝑦 E 𝑋𝑦𝑋))
97, 8bitrd 267 . . . . 5 (𝑋𝐵 → (𝑋 E 𝑦𝑦𝑋))
109abbi1dv 2730 . . . 4 (𝑋𝐵 → {𝑦𝑋 E 𝑦} = 𝑋)
114, 10syl5eq 2656 . . 3 (𝑋𝐵 → ( E “ {𝑋}) = 𝑋)
1211ineq2d 3776 . 2 (𝑋𝐵 → (𝐴 ∩ ( E “ {𝑋})) = (𝐴𝑋))
131, 12syl5eq 2656 1 (𝑋𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ∩ cin 3539  {csn 4125   class class class wbr 4583   E cep 4947  ◡ccnv 5037   “ cima 5041  Rel wrel 5043  Predcpred 5596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597 This theorem is referenced by:  predon  6883  omsinds  6976
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