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Theorem fvtp2g 6056
Description: The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
Assertion
Ref Expression
fvtp2g  |-  ( ( ( B  e.  V  /\  E  e.  W
)  /\  ( A  =/=  B  /\  B  =/= 
C ) )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B
)  =  E )

Proof of Theorem fvtp2g
StepHypRef Expression
1 tprot 4064 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. }
21fveq1i 5804 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 B )  =  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)
3 necom 2670 . . . 4  |-  ( A  =/=  B  <->  B  =/=  A )
4 fvtp1g 6055 . . . . . 6  |-  ( ( ( B  e.  V  /\  E  e.  W
)  /\  ( B  =/=  C  /\  B  =/= 
A ) )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
54expcom 433 . . . . 5  |-  ( ( B  =/=  C  /\  B  =/=  A )  -> 
( ( B  e.  V  /\  E  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B )  =  E ) )
65ancoms 451 . . . 4  |-  ( ( B  =/=  A  /\  B  =/=  C )  -> 
( ( B  e.  V  /\  E  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B )  =  E ) )
73, 6sylanb 470 . . 3  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( ( B  e.  V  /\  E  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B )  =  E ) )
87impcom 428 . 2  |-  ( ( ( B  e.  V  /\  E  e.  W
)  /\  ( A  =/=  B  /\  B  =/= 
C ) )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
92, 8syl5eq 2453 1  |-  ( ( ( B  e.  V  /\  E  e.  W
)  /\  ( A  =/=  B  /\  B  =/= 
C ) )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B
)  =  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840    =/= wne 2596   {ctp 3973   <.cop 3975   ` cfv 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-res 4952  df-iota 5487  df-fun 5525  df-fv 5531
This theorem is referenced by:  fvtp3g  6057  2wlklemB  24856
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