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

Theorem elrn 5112
Description: Membership in a range. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
elrn.1  |-  A  e. 
_V
Assertion
Ref Expression
elrn  |-  ( A  e.  ran  B  <->  E. x  x B A )
Distinct variable groups:    x, A    x, B

Proof of Theorem elrn
StepHypRef Expression
1 elrn.1 . . 3  |-  A  e. 
_V
21elrn2 5111 . 2  |-  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B )
3 df-br 4215 . . 3  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
43exbii 1593 . 2  |-  ( E. x  x B A  <->  E. x <. x ,  A >.  e.  B )
52, 4bitr4i 245 1  |-  ( A  e.  ran  B  <->  E. x  x B A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    e. wcel 1726   _Vcvv 2958   <.cop 3819   class class class wbr 4214   ran crn 4881
This theorem is referenced by:  dmcosseq  5139  rnco  5378  dffo4  5887  fvclss  5982  rntpos  6494  fpwwe2lem11  8517  fpwwe2lem12  8518  fclim  12349  perfdvf  19792  dftr6  25375  dffr5  25378  brsset  25736  dfon3  25739  brtxpsd  25741  dffix2  25752  elsingles  25765  dfrdg4  25797  inisegn0  27120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-cnv 4888  df-dm 4890  df-rn 4891
  Copyright terms: Public domain W3C validator