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Theorem fvtp2 6123
Description: The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
fvtp2.1  |-  B  e. 
_V
fvtp2.4  |-  E  e. 
_V
Assertion
Ref Expression
fvtp2  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B
)  =  E )

Proof of Theorem fvtp2
StepHypRef Expression
1 tprot 4092 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. }
21fveq1i 5878 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 B )  =  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)
3 necom 2693 . . 3  |-  ( A  =/=  B  <->  B  =/=  A )
4 fvtp2.1 . . . . 5  |-  B  e. 
_V
5 fvtp2.4 . . . . 5  |-  E  e. 
_V
64, 5fvtp1 6122 . . . 4  |-  ( ( B  =/=  C  /\  B  =/=  A )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
76ancoms 454 . . 3  |-  ( ( B  =/=  A  /\  B  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
83, 7sylanb 474 . 2  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
92, 8syl5eq 2475 1  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B
)  =  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   _Vcvv 3081   {ctp 4000   <.cop 4002   ` cfv 5597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-res 4861  df-iota 5561  df-fun 5599  df-fv 5605
This theorem is referenced by:  fvtp3  6124  wlkntrllem2  25275  constr3lem5  25361  rabren3dioph  35576  nnsum4primesodd  38602  nnsum4primesoddALTV  38603
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