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Theorem fvtp2g 6041
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 4081 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. }
21fveq1i 5803 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 B )  =  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)
3 necom 2721 . . . 4  |-  ( A  =/=  B  <->  B  =/=  A )
4 fvtp1g 6040 . . . . . 6  |-  ( ( ( B  e.  V  /\  E  e.  W
)  /\  ( B  =/=  C  /\  B  =/= 
A ) )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
54expcom 435 . . . . 5  |-  ( ( B  =/=  C  /\  B  =/=  A )  -> 
( ( B  e.  V  /\  E  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B )  =  E ) )
65ancoms 453 . . . 4  |-  ( ( B  =/=  A  /\  B  =/=  C )  -> 
( ( B  e.  V  /\  E  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B )  =  E ) )
73, 6sylanb 472 . . 3  |-  ( ( A  =/=  B  /\  B  =/=  C )  -> 
( ( B  e.  V  /\  E  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B )  =  E ) )
87impcom 430 . 2  |-  ( ( ( B  e.  V  /\  E  e.  W
)  /\  ( A  =/=  B  /\  B  =/= 
C ) )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  B
)  =  E )
92, 8syl5eq 2507 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 369    = wceq 1370    e. wcel 1758    =/= wne 2648   {ctp 3992   <.cop 3994   ` cfv 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-res 4963  df-iota 5492  df-fun 5531  df-fv 5537
This theorem is referenced by:  fvtp3g  6042  2wlklemB  23626
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