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Theorem eliniseg 4949
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 4946 . . . 4  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  <. B ,  C >.  e.  `' A
) )
3 df-br 3921 . . . 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 4769 . . 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 2727   {csn 3544   <.cop 3547   class class class wbr 3920   `'ccnv 4579   "cima 4583
This theorem is referenced by:  epini  4950  iniseg  4951  dfco2a  5079  isomin  5686  isoini  5687  fnse  6084  infxpenlem  7525  fpwwe2lem8  8139  fpwwe2lem12  8143  fpwwe2lem13  8144  fpwwe2  8145  canth4  8149  canthwelem  8152  pwfseqlem4  8164  fz1isolem  11276  itg1addlem4  18886  elnlfn  22338  elpred  23345  pw2f1ocnv  26296
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601
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