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Theorem f1dom3fv3dif 6186
Description: The function values for a 1-1 function from a set with three different elements are different. (Contributed by AV, 20-Mar-2019.)
Hypotheses
Ref Expression
f1dom3fv3dif.v  |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )
f1dom3fv3dif.n  |-  ( ph  ->  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
f1dom3fv3dif.f  |-  ( ph  ->  F : { A ,  B ,  C } -1-1->
R )
Assertion
Ref Expression
f1dom3fv3dif  |-  ( ph  ->  ( ( F `  A )  =/=  ( F `  B )  /\  ( F `  A
)  =/=  ( F `
 C )  /\  ( F `  B )  =/=  ( F `  C ) ) )

Proof of Theorem f1dom3fv3dif
StepHypRef Expression
1 f1dom3fv3dif.n . . . 4  |-  ( ph  ->  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
21simp1d 1042 . . 3  |-  ( ph  ->  A  =/=  B )
3 f1dom3fv3dif.f . . . . 5  |-  ( ph  ->  F : { A ,  B ,  C } -1-1->
R )
4 eqidd 2472 . . . . . . 7  |-  ( ph  ->  A  =  A )
543mix1d 1205 . . . . . 6  |-  ( ph  ->  ( A  =  A  \/  A  =  B  \/  A  =  C ) )
6 f1dom3fv3dif.v . . . . . . . 8  |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )
76simp1d 1042 . . . . . . 7  |-  ( ph  ->  A  e.  X )
8 eltpg 4005 . . . . . . 7  |-  ( A  e.  X  ->  ( A  e.  { A ,  B ,  C }  <->  ( A  =  A  \/  A  =  B  \/  A  =  C )
) )
97, 8syl 17 . . . . . 6  |-  ( ph  ->  ( A  e.  { A ,  B ,  C }  <->  ( A  =  A  \/  A  =  B  \/  A  =  C ) ) )
105, 9mpbird 240 . . . . 5  |-  ( ph  ->  A  e.  { A ,  B ,  C }
)
11 eqidd 2472 . . . . . . 7  |-  ( ph  ->  B  =  B )
12113mix2d 1206 . . . . . 6  |-  ( ph  ->  ( B  =  A  \/  B  =  B  \/  B  =  C ) )
136simp2d 1043 . . . . . . 7  |-  ( ph  ->  B  e.  Y )
14 eltpg 4005 . . . . . . 7  |-  ( B  e.  Y  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C )
) )
1513, 14syl 17 . . . . . 6  |-  ( ph  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C ) ) )
1612, 15mpbird 240 . . . . 5  |-  ( ph  ->  B  e.  { A ,  B ,  C }
)
17 f1fveq 6181 . . . . 5  |-  ( ( F : { A ,  B ,  C } -1-1->
R  /\  ( A  e.  { A ,  B ,  C }  /\  B  e.  { A ,  B ,  C } ) )  ->  ( ( F `
 A )  =  ( F `  B
)  <->  A  =  B
) )
183, 10, 16, 17syl12anc 1290 . . . 4  |-  ( ph  ->  ( ( F `  A )  =  ( F `  B )  <-> 
A  =  B ) )
1918necon3bid 2687 . . 3  |-  ( ph  ->  ( ( F `  A )  =/=  ( F `  B )  <->  A  =/=  B ) )
202, 19mpbird 240 . 2  |-  ( ph  ->  ( F `  A
)  =/=  ( F `
 B ) )
211simp2d 1043 . . 3  |-  ( ph  ->  A  =/=  C )
226simp3d 1044 . . . . . 6  |-  ( ph  ->  C  e.  Z )
23 tpid3g 4078 . . . . . 6  |-  ( C  e.  Z  ->  C  e.  { A ,  B ,  C } )
2422, 23syl 17 . . . . 5  |-  ( ph  ->  C  e.  { A ,  B ,  C }
)
25 f1fveq 6181 . . . . 5  |-  ( ( F : { A ,  B ,  C } -1-1->
R  /\  ( A  e.  { A ,  B ,  C }  /\  C  e.  { A ,  B ,  C } ) )  ->  ( ( F `
 A )  =  ( F `  C
)  <->  A  =  C
) )
263, 10, 24, 25syl12anc 1290 . . . 4  |-  ( ph  ->  ( ( F `  A )  =  ( F `  C )  <-> 
A  =  C ) )
2726necon3bid 2687 . . 3  |-  ( ph  ->  ( ( F `  A )  =/=  ( F `  C )  <->  A  =/=  C ) )
2821, 27mpbird 240 . 2  |-  ( ph  ->  ( F `  A
)  =/=  ( F `
 C ) )
291simp3d 1044 . . 3  |-  ( ph  ->  B  =/=  C )
30 f1fveq 6181 . . . . 5  |-  ( ( F : { A ,  B ,  C } -1-1->
R  /\  ( B  e.  { A ,  B ,  C }  /\  C  e.  { A ,  B ,  C } ) )  ->  ( ( F `
 B )  =  ( F `  C
)  <->  B  =  C
) )
313, 16, 24, 30syl12anc 1290 . . . 4  |-  ( ph  ->  ( ( F `  B )  =  ( F `  C )  <-> 
B  =  C ) )
3231necon3bid 2687 . . 3  |-  ( ph  ->  ( ( F `  B )  =/=  ( F `  C )  <->  B  =/=  C ) )
3329, 32mpbird 240 . 2  |-  ( ph  ->  ( F `  B
)  =/=  ( F `
 C ) )
3420, 28, 333jca 1210 1  |-  ( ph  ->  ( ( F `  A )  =/=  ( F `  B )  /\  ( F `  A
)  =/=  ( F `
 C )  /\  ( F `  B )  =/=  ( F `  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ w3o 1006    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   {ctp 3963   -1-1->wf1 5586   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fv 5597
This theorem is referenced by:  f1dom3el3dif  6187
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