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Theorem fvtp3g 6043
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
fvtp3g  |-  ( ( ( C  e.  V  /\  F  e.  W
)  /\  ( A  =/=  C  /\  B  =/= 
C ) )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C
)  =  F )

Proof of Theorem fvtp3g
StepHypRef Expression
1 tprot 4056 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. }
21fveq1i 5792 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 C )  =  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)
3 necom 2665 . . . . 5  |-  ( A  =/=  C  <->  C  =/=  A )
4 fvtp2g 6042 . . . . . 6  |-  ( ( ( C  e.  V  /\  F  e.  W
)  /\  ( B  =/=  C  /\  C  =/= 
A ) )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
54expcom 433 . . . . 5  |-  ( ( B  =/=  C  /\  C  =/=  A )  -> 
( ( C  e.  V  /\  F  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C )  =  F ) )
63, 5sylan2b 473 . . . 4  |-  ( ( B  =/=  C  /\  A  =/=  C )  -> 
( ( C  e.  V  /\  F  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C )  =  F ) )
76ancoms 451 . . 3  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( C  e.  V  /\  F  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C )  =  F ) )
87impcom 428 . 2  |-  ( ( ( C  e.  V  /\  F  e.  W
)  /\  ( A  =/=  C  /\  B  =/= 
C ) )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
92, 8syl5eq 2449 1  |-  ( ( ( C  e.  V  /\  F  e.  W
)  /\  ( A  =/=  C  /\  B  =/= 
C ) )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C
)  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836    =/= wne 2591   {ctp 3965   <.cop 3967   ` cfv 5513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-res 4942  df-iota 5477  df-fun 5515  df-fv 5521
This theorem is referenced by:  2wlklemC  24704  2pthon  24750
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