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Theorem predep 5704
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 5678 . 2 Pred( E , 𝐴, 𝑋) = (𝐴 ∩ ( E “ {𝑋}))
2 relcnv 5501 . . . . 5 Rel E
3 relimasn 5486 . . . . 5 (Rel E → ( E “ {𝑋}) = {𝑦𝑋 E 𝑦})
42, 3ax-mp 5 . . . 4 ( E “ {𝑋}) = {𝑦𝑋 E 𝑦}
5 vex 3201 . . . . . . 7 𝑦 ∈ V
6 brcnvg 5301 . . . . . . 7 ((𝑋𝐵𝑦 ∈ V) → (𝑋 E 𝑦𝑦 E 𝑋))
75, 6mpan2 707 . . . . . 6 (𝑋𝐵 → (𝑋 E 𝑦𝑦 E 𝑋))
8 epelg 5028 . . . . . 6 (𝑋𝐵 → (𝑦 E 𝑋𝑦𝑋))
97, 8bitrd 268 . . . . 5 (𝑋𝐵 → (𝑋 E 𝑦𝑦𝑋))
109abbi1dv 2742 . . . 4 (𝑋𝐵 → {𝑦𝑋 E 𝑦} = 𝑋)
114, 10syl5eq 2667 . . 3 (𝑋𝐵 → ( E “ {𝑋}) = 𝑋)
1211ineq2d 3812 . 2 (𝑋𝐵 → (𝐴 ∩ ( E “ {𝑋})) = (𝐴𝑋))
131, 12syl5eq 2667 1 (𝑋𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1482  wcel 1989  {cab 2607  Vcvv 3198  cin 3571  {csn 4175   class class class wbr 4651   E cep 5026  ccnv 5111  cima 5115  Rel wrel 5117  Predcpred 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-eprel 5027  df-xp 5118  df-rel 5119  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678
This theorem is referenced by:  predon  6988  omsinds  7081
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