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Theorem fvtp2 6120
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 4128 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. }
21fveq1i 5873 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 B )  =  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)
3 necom 2736 . . 3  |-  ( A  =/=  B  <->  B  =/=  A )
4 fvtp2.1 . . . . 5  |-  B  e. 
_V
5 fvtp2.4 . . . . 5  |-  E  e. 
_V
64, 5fvtp1 6119 . . . 4  |-  ( ( B  =/=  C  /\  B  =/=  A )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
76ancoms 453 . . 3  |-  ( ( B  =/=  A  /\  B  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
83, 7sylanb 472 . 2  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
92, 8syl5eq 2520 1  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B
)  =  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118   {ctp 4037   <.cop 4039   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-res 5017  df-iota 5557  df-fun 5596  df-fv 5602
This theorem is referenced by:  fvtp3  6121  wlkntrllem2  24385  constr3lem5  24471  rabren3dioph  30677
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