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Theorem f1dom3fv3dif 6183
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 1017 . . 3  |-  ( ph  ->  A  =/=  B )
3 f1dom3fv3dif.f . . . . 5  |-  ( ph  ->  F : { A ,  B ,  C } -1-1->
R )
4 eqidd 2423 . . . . . . 7  |-  ( ph  ->  A  =  A )
543mix1d 1180 . . . . . 6  |-  ( ph  ->  ( A  =  A  \/  A  =  B  \/  A  =  C ) )
6 f1dom3fv3dif.v . . . . . . . 8  |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )
76simp1d 1017 . . . . . . 7  |-  ( ph  ->  A  e.  X )
8 eltpg 4042 . . . . . . 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 235 . . . . 5  |-  ( ph  ->  A  e.  { A ,  B ,  C }
)
11 eqidd 2423 . . . . . . 7  |-  ( ph  ->  B  =  B )
12113mix2d 1181 . . . . . 6  |-  ( ph  ->  ( B  =  A  \/  B  =  B  \/  B  =  C ) )
136simp2d 1018 . . . . . . 7  |-  ( ph  ->  B  e.  Y )
14 eltpg 4042 . . . . . . 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 235 . . . . 5  |-  ( ph  ->  B  e.  { A ,  B ,  C }
)
17 f1fveq 6178 . . . . 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 1262 . . . 4  |-  ( ph  ->  ( ( F `  A )  =  ( F `  B )  <-> 
A  =  B ) )
1918necon3bid 2678 . . 3  |-  ( ph  ->  ( ( F `  A )  =/=  ( F `  B )  <->  A  =/=  B ) )
202, 19mpbird 235 . 2  |-  ( ph  ->  ( F `  A
)  =/=  ( F `
 B ) )
211simp2d 1018 . . 3  |-  ( ph  ->  A  =/=  C )
226simp3d 1019 . . . . . 6  |-  ( ph  ->  C  e.  Z )
23 tpid3g 4115 . . . . . 6  |-  ( C  e.  Z  ->  C  e.  { A ,  B ,  C } )
2422, 23syl 17 . . . . 5  |-  ( ph  ->  C  e.  { A ,  B ,  C }
)
25 f1fveq 6178 . . . . 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 1262 . . . 4  |-  ( ph  ->  ( ( F `  A )  =  ( F `  C )  <-> 
A  =  C ) )
2726necon3bid 2678 . . 3  |-  ( ph  ->  ( ( F `  A )  =/=  ( F `  C )  <->  A  =/=  C ) )
2821, 27mpbird 235 . 2  |-  ( ph  ->  ( F `  A
)  =/=  ( F `
 C ) )
291simp3d 1019 . . 3  |-  ( ph  ->  B  =/=  C )
30 f1fveq 6178 . . . . 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 1262 . . . 4  |-  ( ph  ->  ( ( F `  B )  =  ( F `  C )  <-> 
B  =  C ) )
3231necon3bid 2678 . . 3  |-  ( ph  ->  ( ( F `  B )  =/=  ( F `  C )  <->  B  =/=  C ) )
3329, 32mpbird 235 . 2  |-  ( ph  ->  ( F `  B
)  =/=  ( F `
 C ) )
3420, 28, 333jca 1185 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 187    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   {ctp 4002   -1-1->wf1 5598   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fv 5609
This theorem is referenced by:  f1dom3el3dif  6184
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