MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  predasetex Structured version   Unicode version

Theorem predasetex 5382
Description: The predecessor class exists when  A does. (Contributed by Scott Fenton, 8-Feb-2011.)
Hypothesis
Ref Expression
predasetex.1  |-  A  e. 
_V
Assertion
Ref Expression
predasetex  |-  Pred ( R ,  A ,  X )  e.  _V

Proof of Theorem predasetex
StepHypRef Expression
1 df-pred 5367 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
2 predasetex.1 . . 3  |-  A  e. 
_V
32inex1 4535 . 2  |-  ( A  i^i  ( `' R " { X } ) )  e.  _V
41, 3eqeltri 2486 1  |-  Pred ( R ,  A ,  X )  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1842   _Vcvv 3059    i^i cin 3413   {csn 3972   `'ccnv 4822   "cima 4826   Predcpred 5366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3061  df-in 3421  df-pred 5367
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator