MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fntp Structured version   Unicode version

Theorem fntp 5510
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 5506 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  Fun  {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } )
84, 5, 6dmtpop 5349 . . 3  |-  dom  { <. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. }  =  { A ,  B ,  C }
98a1i 11 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  dom  {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. }  =  { A ,  B ,  C }
)
10 df-fn 5460 . 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 647 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 937    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958   {ctp 3818   <.cop 3819   dom cdm 4881   Fun wfun 5451    Fn wfn 5452
This theorem is referenced by:  rabren3dioph  26890
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-tp 3824  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  df-fn 5460
  Copyright terms: Public domain W3C validator