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Theorem predep 6167
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 6141 . 2 Pred( E , 𝐴, 𝑋) = (𝐴 ∩ ( E “ {𝑋}))
2 relcnv 5960 . . . . 5 Rel E
3 relimasn 5945 . . . . 5 (Rel E → ( E “ {𝑋}) = {𝑦𝑋 E 𝑦})
42, 3ax-mp 5 . . . 4 ( E “ {𝑋}) = {𝑦𝑋 E 𝑦}
5 brcnvg 5743 . . . . . . 7 ((𝑋𝐵𝑦 ∈ V) → (𝑋 E 𝑦𝑦 E 𝑋))
65elvd 3499 . . . . . 6 (𝑋𝐵 → (𝑋 E 𝑦𝑦 E 𝑋))
7 epelg 5459 . . . . . 6 (𝑋𝐵 → (𝑦 E 𝑋𝑦𝑋))
86, 7bitrd 281 . . . . 5 (𝑋𝐵 → (𝑋 E 𝑦𝑦𝑋))
98abbi1dv 2950 . . . 4 (𝑋𝐵 → {𝑦𝑋 E 𝑦} = 𝑋)
104, 9syl5eq 2866 . . 3 (𝑋𝐵 → ( E “ {𝑋}) = 𝑋)
1110ineq2d 4187 . 2 (𝑋𝐵 → (𝐴 ∩ ( E “ {𝑋})) = (𝐴𝑋))
121, 11syl5eq 2866 1 (𝑋𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1530  wcel 2107  {cab 2797  Vcvv 3493  cin 3933  {csn 4559   class class class wbr 5057   E cep 5457  ccnv 5547  cima 5551  Rel wrel 5553  Predcpred 6140
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-eprel 5458  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141
This theorem is referenced by:  predon  7498  omsinds  7592
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