MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvtp1g Structured version   Unicode version

Theorem fvtp1g 6109
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
fvtp1g  |-  ( ( ( A  e.  V  /\  D  e.  W
)  /\  ( A  =/=  B  /\  A  =/= 
C ) )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  A
)  =  D )

Proof of Theorem fvtp1g
StepHypRef Expression
1 df-tp 4032 . . 3  |-  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  =  ( { <. A ,  D >. ,  <. B ,  E >. }  u.  { <. C ,  F >. } )
21fveq1i 5865 . 2  |-  ( {
<. A ,  D >. , 
<. B ,  E >. , 
<. C ,  F >. } `
 A )  =  ( ( { <. A ,  D >. ,  <. B ,  E >. }  u.  {
<. C ,  F >. } ) `  A )
3 necom 2736 . . . . 5  |-  ( A  =/=  C  <->  C  =/=  A )
4 fvunsn 6091 . . . . 5  |-  ( C  =/=  A  ->  (
( { <. A ,  D >. ,  <. B ,  E >. }  u.  { <. C ,  F >. } ) `  A )  =  ( { <. A ,  D >. ,  <. B ,  E >. } `  A ) )
53, 4sylbi 195 . . . 4  |-  ( A  =/=  C  ->  (
( { <. A ,  D >. ,  <. B ,  E >. }  u.  { <. C ,  F >. } ) `  A )  =  ( { <. A ,  D >. ,  <. B ,  E >. } `  A ) )
65ad2antll 728 . . 3  |-  ( ( ( A  e.  V  /\  D  e.  W
)  /\  ( A  =/=  B  /\  A  =/= 
C ) )  -> 
( ( { <. A ,  D >. ,  <. B ,  E >. }  u.  {
<. C ,  F >. } ) `  A )  =  ( { <. A ,  D >. ,  <. B ,  E >. } `  A ) )
7 fvpr1g 6104 . . . . 5  |-  ( ( A  e.  V  /\  D  e.  W  /\  A  =/=  B )  -> 
( { <. A ,  D >. ,  <. B ,  E >. } `  A
)  =  D )
873expa 1196 . . . 4  |-  ( ( ( A  e.  V  /\  D  e.  W
)  /\  A  =/=  B )  ->  ( { <. A ,  D >. , 
<. B ,  E >. } `
 A )  =  D )
98adantrr 716 . . 3  |-  ( ( ( A  e.  V  /\  D  e.  W
)  /\  ( A  =/=  B  /\  A  =/= 
C ) )  -> 
( { <. A ,  D >. ,  <. B ,  E >. } `  A
)  =  D )
106, 9eqtrd 2508 . 2  |-  ( ( ( A  e.  V  /\  D  e.  W
)  /\  ( A  =/=  B  /\  A  =/= 
C ) )  -> 
( ( { <. A ,  D >. ,  <. B ,  E >. }  u.  {
<. C ,  F >. } ) `  A )  =  D )
112, 10syl5eq 2520 1  |-  ( ( ( A  e.  V  /\  D  e.  W
)  /\  ( A  =/=  B  /\  A  =/= 
C ) )  -> 
( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  A
)  =  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    u. cun 3474   {csn 4027   {cpr 4029   {ctp 4031   <.cop 4033   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5549  df-fun 5588  df-fv 5594
This theorem is referenced by:  fvtp2g  6110  2wlklemA  24232  2pthon  24280
  Copyright terms: Public domain W3C validator