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Theorem isof1o 5472
Description: An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
Assertion
Ref Expression
isof1o  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)

Proof of Theorem isof1o
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-isom 4935 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
21simplbi 268 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wral 2349   class class class wbr 3787   -1-1-onto->wf1o 4925   ` cfv 4926    Isom wiso 4927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104
This theorem depends on definitions:  df-bi 115  df-isom 4935
This theorem is referenced by:  isocnv2  5477  isores1  5479  isoini  5482  isoini2  5483  isoselem  5484  isose  5485  isopolem  5486  isosolem  5488  smoiso  5945  isotilem  6468  supisolem  6470  supisoex  6471  supisoti  6472  ordiso2  6495
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