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Theorem eliniseg 5195
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 5192 . . . 4  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  <. B ,  C >.  e.  `' A
) )
3 df-br 4290 . . . 4  |-  ( B `' A C  <->  <. B ,  C >.  e.  `' A
)
42, 3syl6bbr 263 . . 3  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  B `' A C ) )
5 brcnvg 5016 . . 3  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( B `' A C 
<->  C A B ) )
64, 5bitrd 253 . 2  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  C A B ) )
71, 6mpan2 666 1  |-  ( B  e.  V  ->  ( C  e.  ( `' A " { B }
)  <->  C A B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1761   _Vcvv 2970   {csn 3874   <.cop 3880   class class class wbr 4289   `'ccnv 4835   "cima 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-xp 4842  df-cnv 4844  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849
This theorem is referenced by:  epini  5196  iniseg  5197  dfco2a  5335  isomin  6025  isoini  6026  fnse  6688  infxpenlem  8176  fpwwe2lem8  8800  fpwwe2lem12  8804  fpwwe2lem13  8805  fpwwe2  8806  canth4  8810  canthwelem  8813  pwfseqlem4  8825  fz1isolem  12210  itg1addlem4  21077  elnlfn  25251  elpred  27551  pw2f1ocnv  29295
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