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Theorem fvtp3 6102
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 4069 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. }
21fveq1i 5852 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 C )  =  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)
3 necom 2674 . . . 4  |-  ( A  =/=  C  <->  C  =/=  A )
4 fvtp3.1 . . . . 5  |-  C  e. 
_V
5 fvtp3.4 . . . . 5  |-  F  e. 
_V
64, 5fvtp2 6101 . . . 4  |-  ( ( B  =/=  C  /\  C  =/=  A )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
73, 6sylan2b 475 . . 3  |-  ( ( B  =/=  C  /\  A  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
87ancoms 453 . 2  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
92, 8syl5eq 2457 1  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C
)  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600   _Vcvv 3061   {ctp 3978   <.cop 3980   ` cfv 5571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-res 4837  df-iota 5535  df-fun 5573  df-fv 5579
This theorem is referenced by:  wlkntrllem2  24991  constr3lem5  25077  rabren3dioph  35123  nnsum4primesodd  37857  nnsum4primesoddALTV  37858
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