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Theorem funpr 5505
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 443 . 2  |-  ( A  e.  _V  /\  B  e.  _V )
4 funpr.3 . . 3  |-  C  e. 
_V
5 funpr.4 . . 3  |-  D  e. 
_V
64, 5pm3.2i 443 . 2  |-  ( C  e.  _V  /\  D  e.  _V )
7 funprg 5503 . 2  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  ( C  e.  _V  /\  D  e.  _V )  /\  A  =/=  B
)  ->  Fun  { <. A ,  C >. ,  <. B ,  D >. } )
83, 6, 7mp3an12 1270 1  |-  ( A  =/=  B  ->  Fun  {
<. A ,  C >. , 
<. B ,  D >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726    =/= wne 2601   _Vcvv 2958   {cpr 3817   <.cop 3819   Fun wfun 5451
This theorem is referenced by:  funtp  5506  fpr  5917  fnpr  5953  fnprOLD  5954  1sdom  7314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-fun 5459
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