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Theorem fntp 5322
Description: A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
fntp.1  |-  A  e. 
_V
fntp.2  |-  B  e. 
_V
fntp.3  |-  C  e. 
_V
fntp.4  |-  D  e. 
_V
fntp.5  |-  E  e. 
_V
fntp.6  |-  F  e. 
_V
Assertion
Ref Expression
fntp  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  Fn  { A ,  B ,  C } )

Proof of Theorem fntp
StepHypRef Expression
1 fntp.1 . . 3  |-  A  e. 
_V
2 fntp.2 . . 3  |-  B  e. 
_V
3 fntp.3 . . 3  |-  C  e. 
_V
4 fntp.4 . . 3  |-  D  e. 
_V
5 fntp.5 . . 3  |-  E  e. 
_V
6 fntp.6 . . 3  |-  F  e. 
_V
71, 2, 3, 4, 5, 6funtp 5319 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  Fun  {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } )
84, 5, 6dmtpop 5165 . . 3  |-  dom  { <. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. }  =  { A ,  B ,  C }
98a1i 10 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  dom  {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. }  =  { A ,  B ,  C }
)
10 df-fn 5274 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. }  Fn  { A ,  B ,  C }  <->  ( Fun  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  /\  dom  {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. }  =  { A ,  B ,  C }
) )
117, 9, 10sylanbrc 645 1  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  Fn  { A ,  B ,  C } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   {ctp 3655   <.cop 3656   dom cdm 4705   Fun wfun 5265    Fn wfn 5266
This theorem is referenced by:  rabren3dioph  27001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274
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