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Theorem predep 5849
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 5823 . 2 Pred( E , 𝐴, 𝑋) = (𝐴 ∩ ( E “ {𝑋}))
2 relcnv 5644 . . . . 5 Rel E
3 relimasn 5629 . . . . 5 (Rel E → ( E “ {𝑋}) = {𝑦𝑋 E 𝑦})
42, 3ax-mp 5 . . . 4 ( E “ {𝑋}) = {𝑦𝑋 E 𝑦}
5 vex 3354 . . . . . . 7 𝑦 ∈ V
6 brcnvg 5441 . . . . . . 7 ((𝑋𝐵𝑦 ∈ V) → (𝑋 E 𝑦𝑦 E 𝑋))
75, 6mpan2 671 . . . . . 6 (𝑋𝐵 → (𝑋 E 𝑦𝑦 E 𝑋))
8 epelg 5163 . . . . . 6 (𝑋𝐵 → (𝑦 E 𝑋𝑦𝑋))
97, 8bitrd 268 . . . . 5 (𝑋𝐵 → (𝑋 E 𝑦𝑦𝑋))
109abbi1dv 2892 . . . 4 (𝑋𝐵 → {𝑦𝑋 E 𝑦} = 𝑋)
114, 10syl5eq 2817 . . 3 (𝑋𝐵 → ( E “ {𝑋}) = 𝑋)
1211ineq2d 3965 . 2 (𝑋𝐵 → (𝐴 ∩ ( E “ {𝑋})) = (𝐴𝑋))
131, 12syl5eq 2817 1 (𝑋𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  {cab 2757  Vcvv 3351  cin 3722  {csn 4316   class class class wbr 4786   E cep 5161  ccnv 5248  cima 5252  Rel wrel 5254  Predcpred 5822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-eprel 5162  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823
This theorem is referenced by:  predon  7138  omsinds  7231
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