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Theorem funpr 5406
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 441 . 2  |-  ( A  e.  _V  /\  B  e.  _V )
4 funpr.3 . . 3  |-  C  e. 
_V
5 funpr.4 . . 3  |-  D  e. 
_V
64, 5pm3.2i 441 . 2  |-  ( C  e.  _V  /\  D  e.  _V )
7 funprg 5404 . 2  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  ( C  e.  _V  /\  D  e.  _V )  /\  A  =/=  B
)  ->  Fun  { <. A ,  C >. ,  <. B ,  D >. } )
83, 6, 7mp3an12 1268 1  |-  ( A  =/=  B  ->  Fun  {
<. A ,  C >. , 
<. B ,  D >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1715    =/= wne 2529   _Vcvv 2873   {cpr 3730   <.cop 3732   Fun wfun 5352
This theorem is referenced by:  funtp  5407  fpr  5815  fnpr  5850  fnprOLD  5851  1sdom  7208  prfunOLD  25954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-fun 5360
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