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

Theorem iotacl 5404
Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5381). If you have a bounded iota-based definition, riotacl2 6526 may be useful.

(Contributed by Andrew Salmon, 1-Aug-2011.)

Assertion
Ref Expression
iotacl  |-  ( E! x ph  ->  ( iota x ph )  e. 
{ x  |  ph } )

Proof of Theorem iotacl
StepHypRef Expression
1 iota4 5399 . 2  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
2 df-sbc 3126 . 2  |-  ( [. ( iota x ph )  /  x ]. ph  <->  ( iota x ph )  e.  {
x  |  ph }
)
31, 2sylib 189 1  |-  ( E! x ph  ->  ( iota x ph )  e. 
{ x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   E!weu 2258   {cab 2394   [.wsbc 3125   iotacio 5379
This theorem is referenced by:  opiota  6498  riotacl2  6526  eroprf  6965  iunfictbso  7955  isf32lem9  8201  psgnvali  27303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rex 2676  df-v 2922  df-sbc 3126  df-un 3289  df-sn 3784  df-pr 3785  df-uni 3980  df-iota 5381
  Copyright terms: Public domain W3C validator