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Theorem bnj1422 28381
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1422.1  |-  ( ph  ->  Fun  A )
bnj1422.2  |-  ( ph  ->  dom  A  =  B )
Assertion
Ref Expression
bnj1422  |-  ( ph  ->  A  Fn  B )

Proof of Theorem bnj1422
StepHypRef Expression
1 bnj1422.1 . 2  |-  ( ph  ->  Fun  A )
2 bnj1422.2 . 2  |-  ( ph  ->  dom  A  =  B )
3 df-fn 5295 . 2  |-  ( A  Fn  B  <->  ( Fun  A  /\  dom  A  =  B ) )
41, 2, 3sylanbrc 645 1  |-  ( ph  ->  A  Fn  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633   dom cdm 4726   Fun wfun 5286    Fn wfn 5287
This theorem is referenced by:  bnj150  28419  bnj535  28433  bnj1312  28599  bnj60  28603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-fn 5295
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