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Theorem fntp 5116
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 5112 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  Fun  {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } )
84, 5, 6dmtpop 4950 . . 3  |-  dom  { <. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. }  =  { A ,  B ,  C }
98a1i 9 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  dom  {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. }  =  { A ,  B ,  C }
)
10 df-fn 5062 . 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 411 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 930    = wceq 1299    e. wcel 1448    =/= wne 2267   _Vcvv 2641   {ctp 3476   <.cop 3477   dom cdm 4477   Fun wfun 5053    Fn wfn 5054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-tp 3482  df-op 3483  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-fun 5061  df-fn 5062
This theorem is referenced by: (None)
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