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

Theorem eliniseg 5224
Description: Membership in an initial segment. The idiom  ( `' A " { B } ), meaning  { x  |  x A B }, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
eliniseg.1  |-  C  e. 
_V
Assertion
Ref Expression
eliniseg  |-  ( B  e.  V  ->  ( C  e.  ( `' A " { B }
)  <->  C A B ) )

Proof of Theorem eliniseg
StepHypRef Expression
1 eliniseg.1 . 2  |-  C  e. 
_V
2 elimasng 5221 . . . 4  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  <. B ,  C >.  e.  `' A
) )
3 df-br 4205 . . . 4  |-  ( B `' A C  <->  <. B ,  C >.  e.  `' A
)
42, 3syl6bbr 255 . . 3  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  B `' A C ) )
5 brcnvg 5044 . . 3  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( B `' A C 
<->  C A B ) )
64, 5bitrd 245 . 2  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  C A B ) )
71, 6mpan2 653 1  |-  ( B  e.  V  ->  ( C  e.  ( `' A " { B }
)  <->  C A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   _Vcvv 2948   {csn 3806   <.cop 3809   class class class wbr 4204   `'ccnv 4868   "cima 4872
This theorem is referenced by:  epini  5225  iniseg  5226  dfco2a  5361  isomin  6048  isoini  6049  fnse  6454  infxpenlem  7884  fpwwe2lem8  8501  fpwwe2lem12  8505  fpwwe2lem13  8506  fpwwe2  8507  canth4  8511  canthwelem  8514  pwfseqlem4  8526  fz1isolem  11698  itg1addlem4  19579  elnlfn  23419  elpred  25432  pw2f1ocnv  27045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-cnv 4877  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882
  Copyright terms: Public domain W3C validator