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Theorem fvtp3g 5936
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 3891 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. }
21fveq1i 5721 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 C )  =  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)
3 necom 2679 . . . . 5  |-  ( A  =/=  C  <->  C  =/=  A )
4 fvtp2g 5935 . . . . . 6  |-  ( ( ( C  e.  V  /\  F  e.  W
)  /\  ( B  =/=  C  /\  C  =/= 
A ) )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
54expcom 425 . . . . 5  |-  ( ( B  =/=  C  /\  C  =/=  A )  -> 
( ( C  e.  V  /\  F  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C )  =  F ) )
63, 5sylan2b 462 . . . 4  |-  ( ( B  =/=  C  /\  A  =/=  C )  -> 
( ( C  e.  V  /\  F  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C )  =  F ) )
76ancoms 440 . . 3  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( C  e.  V  /\  F  e.  W )  ->  ( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C )  =  F ) )
87impcom 420 . 2  |-  ( ( ( C  e.  V  /\  F  e.  W
)  /\  ( A  =/=  C  /\  B  =/= 
C ) )  -> 
( { <. B ,  E >. ,  <. C ,  F >. ,  <. A ,  D >. } `  C
)  =  F )
92, 8syl5eq 2479 1  |-  ( ( ( C  e.  V  /\  F  e.  W
)  /\  ( A  =/=  C  /\  B  =/= 
C ) )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C
)  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   {ctp 3808   <.cop 3809   ` cfv 5446
This theorem is referenced by:  2wlklemC  21548  2pthon  21594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454
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