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

Theorem eldm 4704
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
eldm.1  |-  A  e. 
_V
Assertion
Ref Expression
eldm  |-  ( A  e.  dom  B  <->  E. y  A B y )
Distinct variable groups:    y, A    y, B

Proof of Theorem eldm
StepHypRef Expression
1 eldm.1 . 2  |-  A  e. 
_V
2 eldmg 4702 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
31, 2ax-mp 5 1  |-  ( A  e.  dom  B  <->  E. y  A B y )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   E.wex 1451    e. wcel 1463   _Vcvv 2658   class class class wbr 3897   dom cdm 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-dm 4517
This theorem is referenced by:  dmi  4722  dmcoss  4776  dmcosseq  4778  dminss  4921  dmsnm  4972  dffun7  5118  dffun8  5119  fnres  5207  fndmdif  5491  reldmtpos  6116  dmtpos  6119  tfrexlem  6197
  Copyright terms: Public domain W3C validator