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

Theorem iotacl 5345
Description: Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5322). If you have a bounded iota-based definition, riotacl2 6460 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 5340 . 2  |-  ( E! x ph  ->  [. ( iota x ph )  /  x ]. ph )
2 df-sbc 3078 . 2  |-  ( [. ( iota x ph )  /  x ]. ph  <->  ( iota x ph )  e.  {
x  |  ph }
)
31, 2sylib 188 1  |-  ( E! x ph  ->  ( iota x ph )  e. 
{ x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1715   E!weu 2217   {cab 2352   [.wsbc 3077   iotacio 5320
This theorem is referenced by:  opiota  6432  riotacl2  6460  eroprf  6899  iunfictbso  7888  isf32lem9  8134  psgnvali  26937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rex 2634  df-v 2875  df-sbc 3078  df-un 3243  df-sn 3735  df-pr 3736  df-uni 3930  df-iota 5322
  Copyright terms: Public domain W3C validator