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Theorem fntp 5269
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 5265 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  Fun  {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } )
84, 5, 6dmtpop 5100 . . 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 5215 . 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 417 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 978    = wceq 1353    e. wcel 2148    =/= wne 2347   _Vcvv 2737   {ctp 3593   <.cop 3594   dom cdm 4623   Fun wfun 5206    Fn wfn 5207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-tp 3599  df-op 3600  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-fun 5214  df-fn 5215
This theorem is referenced by: (None)
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