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Theorem funpr 5621
Description: A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
Hypotheses
Ref Expression
funpr.1  |-  A  e. 
_V
funpr.2  |-  B  e. 
_V
funpr.3  |-  C  e. 
_V
funpr.4  |-  D  e. 
_V
Assertion
Ref Expression
funpr  |-  ( A  =/=  B  ->  Fun  {
<. A ,  C >. , 
<. B ,  D >. } )

Proof of Theorem funpr
StepHypRef Expression
1 funpr.1 . . 3  |-  A  e. 
_V
2 funpr.2 . . 3  |-  B  e. 
_V
31, 2pm3.2i 453 . 2  |-  ( A  e.  _V  /\  B  e.  _V )
4 funpr.3 . . 3  |-  C  e. 
_V
5 funpr.4 . . 3  |-  D  e. 
_V
64, 5pm3.2i 453 . 2  |-  ( C  e.  _V  /\  D  e.  _V )
7 funprg 5619 . 2  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  ( C  e.  _V  /\  D  e.  _V )  /\  A  =/=  B
)  ->  Fun  { <. A ,  C >. ,  <. B ,  D >. } )
83, 6, 7mp3an12 1312 1  |-  ( A  =/=  B  ->  Fun  {
<. A ,  C >. , 
<. B ,  D >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823    =/= wne 2649   _Vcvv 3106   {cpr 4018   <.cop 4022   Fun wfun 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-fun 5572
This theorem is referenced by:  funtp  5622  fpr  6055  fnprb  6106  1sdom  7715
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