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Theorem bnj564 32755
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj564.17  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
Assertion
Ref Expression
bnj564  |-  ( ta 
->  dom  f  =  m )

Proof of Theorem bnj564
StepHypRef Expression
1 bnj564.17 . . 3  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
21simp1bi 1006 . 2  |-  ( ta 
->  f  Fn  m
)
3 fndm 5671 . 2  |-  ( f  Fn  m  ->  dom  f  =  m )
42, 3syl 16 1  |-  ( ta 
->  dom  f  =  m )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 968    = wceq 1374   dom cdm 4992    Fn wfn 5574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 970  df-fn 5582
This theorem is referenced by:  bnj570  32917  bnj916  32945
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