Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1422 Structured version   Unicode version

Theorem bnj1422 29146
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 5449 . 2  |-  ( A  Fn  B  <->  ( Fun  A  /\  dom  A  =  B ) )
41, 2, 3sylanbrc 646 1  |-  ( ph  ->  A  Fn  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   dom cdm 4870   Fun wfun 5440    Fn wfn 5441
This theorem is referenced by:  bnj150  29184  bnj535  29198  bnj1312  29364  bnj60  29368
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 178  df-an 361  df-fn 5449
  Copyright terms: Public domain W3C validator