MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  predep Structured version   Visualization version   GIF version

Theorem predep 5893
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 5867 . 2 Pred( E , 𝐴, 𝑋) = (𝐴 ∩ ( E “ {𝑋}))
2 relcnv 5687 . . . . 5 Rel E
3 relimasn 5672 . . . . 5 (Rel E → ( E “ {𝑋}) = {𝑦𝑋 E 𝑦})
42, 3ax-mp 5 . . . 4 ( E “ {𝑋}) = {𝑦𝑋 E 𝑦}
5 brcnvg 5473 . . . . . . 7 ((𝑋𝐵𝑦 ∈ V) → (𝑋 E 𝑦𝑦 E 𝑋))
65elvd 3355 . . . . . 6 (𝑋𝐵 → (𝑋 E 𝑦𝑦 E 𝑋))
7 epelg 5193 . . . . . 6 (𝑋𝐵 → (𝑦 E 𝑋𝑦𝑋))
86, 7bitrd 270 . . . . 5 (𝑋𝐵 → (𝑋 E 𝑦𝑦𝑋))
98abbi1dv 2886 . . . 4 (𝑋𝐵 → {𝑦𝑋 E 𝑦} = 𝑋)
104, 9syl5eq 2811 . . 3 (𝑋𝐵 → ( E “ {𝑋}) = 𝑋)
1110ineq2d 3978 . 2 (𝑋𝐵 → (𝐴 ∩ ( E “ {𝑋})) = (𝐴𝑋))
121, 11syl5eq 2811 1 (𝑋𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1652  wcel 2155  {cab 2751  Vcvv 3350  cin 3733  {csn 4336   class class class wbr 4811   E cep 5191  ccnv 5278  cima 5282  Rel wrel 5284  Predcpred 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-eprel 5192  df-xp 5285  df-rel 5286  df-cnv 5287  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867
This theorem is referenced by:  predon  7193  omsinds  7286
  Copyright terms: Public domain W3C validator