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Theorem eliniseg 4995
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 4992 . . . 4  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  <. B ,  C >.  e.  `' A
) )
3 df-br 3964 . . . 4  |-  ( B `' A C  <->  <. B ,  C >.  e.  `' A
)
42, 3syl6bbr 256 . . 3  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  B `' A C ) )
5 brcnvg 4815 . . 3  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( B `' A C 
<->  C A B ) )
64, 5bitrd 246 . 2  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  C A B ) )
71, 6mpan2 655 1  |-  ( B  e.  V  ->  ( C  e.  ( `' A " { B }
)  <->  C A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1621   _Vcvv 2740   {csn 3581   <.cop 3584   class class class wbr 3963   `'ccnv 4625   "cima 4629
This theorem is referenced by:  epini  4996  iniseg  4997  dfco2a  5125  isomin  5733  isoini  5734  fnse  6131  infxpenlem  7574  fpwwe2lem8  8192  fpwwe2lem12  8196  fpwwe2lem13  8197  fpwwe2  8198  canth4  8202  canthwelem  8205  pwfseqlem4  8217  fz1isolem  11329  itg1addlem4  18981  elnlfn  22433  elpred  23511  pw2f1ocnv  26462
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-xp 4640  df-cnv 4642  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647
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