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Theorem fvtp3 5765
Description: The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
fvtp3.1  |-  C  e. 
_V
fvtp3.4  |-  F  e. 
_V
Assertion
Ref Expression
fvtp3  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C
)  =  F )

Proof of Theorem fvtp3
StepHypRef Expression
1 tprot 3756 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. }
21fveq1i 5564 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 C )  =  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)
3 necom 2560 . . . 4  |-  ( A  =/=  C  <->  C  =/=  A )
4 fvtp3.1 . . . . 5  |-  C  e. 
_V
5 fvtp3.4 . . . . 5  |-  F  e. 
_V
64, 5fvtp2 5764 . . . 4  |-  ( ( B  =/=  C  /\  C  =/=  A )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
73, 6sylan2b 461 . . 3  |-  ( ( B  =/=  C  /\  A  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
87ancoms 439 . 2  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
92, 8syl5eq 2360 1  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C
)  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479   _Vcvv 2822   {ctp 3676   <.cop 3677   ` cfv 5292
This theorem is referenced by:  rabren3dioph  26046  wlkntrllem4  27464  constr3lem5  27532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-res 4738  df-iota 5256  df-fun 5294  df-fv 5300
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