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Theorem fvtp3 5934
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 3892 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. }
21fveq1i 5722 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 C )  =  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)
3 necom 2680 . . . 4  |-  ( A  =/=  C  <->  C  =/=  A )
4 fvtp3.1 . . . . 5  |-  C  e. 
_V
5 fvtp3.4 . . . . 5  |-  F  e. 
_V
64, 5fvtp2 5933 . . . 4  |-  ( ( B  =/=  C  /\  C  =/=  A )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
73, 6sylan2b 462 . . 3  |-  ( ( B  =/=  C  /\  A  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
87ancoms 440 . 2  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
92, 8syl5eq 2480 1  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C
)  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2949   {ctp 3809   <.cop 3810   ` cfv 5447
This theorem is referenced by:  wlkntrllem2  21553  constr3lem5  21628  rabren3dioph  26868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-res 4883  df-iota 5411  df-fun 5449  df-fv 5455
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