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Theorem fvtp2 5872
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 3835 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. }
21fveq1i 5662 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 B )  =  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)
3 necom 2624 . . 3  |-  ( A  =/=  B  <->  B  =/=  A )
4 fvtp2.1 . . . . 5  |-  B  e. 
_V
5 fvtp2.4 . . . . 5  |-  E  e. 
_V
64, 5fvtp1 5871 . . . 4  |-  ( ( B  =/=  C  /\  B  =/=  A )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
76ancoms 440 . . 3  |-  ( ( B  =/=  A  /\  B  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
83, 7sylanb 459 . 2  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
92, 8syl5eq 2424 1  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B
)  =  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   _Vcvv 2892   {ctp 3752   <.cop 3753   ` cfv 5387
This theorem is referenced by:  fvtp3  5873  wlkntrllem4  21409  constr3lem5  21476  rabren3dioph  26560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-res 4823  df-iota 5351  df-fun 5389  df-fv 5395
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