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Theorem fvtp2 5725
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 3722 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. }
21fveq1i 5526 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 B )  =  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)
3 necom 2527 . . 3  |-  ( A  =/=  B  <->  B  =/=  A )
4 fvtp2.1 . . . . 5  |-  B  e. 
_V
5 fvtp2.4 . . . . 5  |-  E  e. 
_V
64, 5fvtp1 5724 . . . 4  |-  ( ( B  =/=  C  /\  B  =/=  A )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
76ancoms 439 . . 3  |-  ( ( B  =/=  A  /\  B  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
83, 7sylanb 458 . 2  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
92, 8syl5eq 2327 1  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B
)  =  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   {ctp 3642   <.cop 3643   ` cfv 5255
This theorem is referenced by:  fvtp3  5726  rabren3dioph  26898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263
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