ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eliniseg Unicode version

Theorem eliniseg 4725
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 4723 . . . 4  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  <. B ,  C >.  e.  `' A
) )
3 df-br 3794 . . . 4  |-  ( B `' A C  <->  <. B ,  C >.  e.  `' A
)
42, 3syl6bbr 196 . . 3  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  B `' A C ) )
5 brcnvg 4544 . . 3  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( B `' A C 
<->  C A B ) )
64, 5bitrd 186 . 2  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( C  e.  ( `' A " { B } )  <->  C A B ) )
71, 6mpan2 416 1  |-  ( B  e.  V  ->  ( C  e.  ( `' A " { B }
)  <->  C A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   _Vcvv 2602   {csn 3406   <.cop 3409   class class class wbr 3793   `'ccnv 4370   "cima 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384
This theorem is referenced by:  epini  4726  iniseg  4727  dfco2a  4851  isoini  5488
  Copyright terms: Public domain W3C validator