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

Theorem fntp 5470
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 5466 . 2  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  Fun  {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } )
84, 5, 6dmtpop 5309 . . 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 5420 . 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 646 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 936    = wceq 1649    e. wcel 1721    =/= wne 2571   _Vcvv 2920   {ctp 3780   <.cop 3781   dom cdm 4841   Fun wfun 5411    Fn wfn 5412
This theorem is referenced by:  rabren3dioph  26770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-fun 5419  df-fn 5420
  Copyright terms: Public domain W3C validator