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Theorem f1dom3el3dif 6187
Description: The range of a 1-1 function from a set with three different elements has (at least) three different elements. (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
f1dom3el3dif  |-  ( ph  ->  E. x  e.  R  E. y  e.  R  E. z  e.  R  ( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z
) )
Distinct variable groups:    x, A, y, z    x, B, y, z    x, C, y, z    x, F, y, z    x, R, y, z
Allowed substitution hints:    ph( x, y, z)    X( x, y, z)    Y( x, y, z)    Z( x, y, z)

Proof of Theorem f1dom3el3dif
StepHypRef Expression
1 f1dom3fv3dif.f . . 3  |-  ( ph  ->  F : { A ,  B ,  C } -1-1->
R )
2 f1f 5792 . . . 4  |-  ( F : { A ,  B ,  C } -1-1->
R  ->  F : { A ,  B ,  C } --> R )
3 simpr 468 . . . . . . 7  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  ->  F : { A ,  B ,  C } --> R )
4 eqidd 2472 . . . . . . . . . 10  |-  ( ph  ->  A  =  A )
543mix1d 1205 . . . . . . . . 9  |-  ( ph  ->  ( A  =  A  \/  A  =  B  \/  A  =  C ) )
6 f1dom3fv3dif.v . . . . . . . . . . 11  |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )
76simp1d 1042 . . . . . . . . . 10  |-  ( ph  ->  A  e.  X )
8 eltpg 4005 . . . . . . . . . 10  |-  ( A  e.  X  ->  ( A  e.  { A ,  B ,  C }  <->  ( A  =  A  \/  A  =  B  \/  A  =  C )
) )
97, 8syl 17 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  { A ,  B ,  C }  <->  ( A  =  A  \/  A  =  B  \/  A  =  C ) ) )
105, 9mpbird 240 . . . . . . . 8  |-  ( ph  ->  A  e.  { A ,  B ,  C }
)
1110adantr 472 . . . . . . 7  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  ->  A  e.  { A ,  B ,  C }
)
123, 11ffvelrnd 6038 . . . . . 6  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  -> 
( F `  A
)  e.  R )
13 eqidd 2472 . . . . . . . . . 10  |-  ( ph  ->  B  =  B )
14133mix2d 1206 . . . . . . . . 9  |-  ( ph  ->  ( B  =  A  \/  B  =  B  \/  B  =  C ) )
156simp2d 1043 . . . . . . . . . 10  |-  ( ph  ->  B  e.  Y )
16 eltpg 4005 . . . . . . . . . 10  |-  ( B  e.  Y  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C )
) )
1715, 16syl 17 . . . . . . . . 9  |-  ( ph  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C ) ) )
1814, 17mpbird 240 . . . . . . . 8  |-  ( ph  ->  B  e.  { A ,  B ,  C }
)
1918adantr 472 . . . . . . 7  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  ->  B  e.  { A ,  B ,  C }
)
203, 19ffvelrnd 6038 . . . . . 6  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  -> 
( F `  B
)  e.  R )
216simp3d 1044 . . . . . . . . 9  |-  ( ph  ->  C  e.  Z )
22 tpid3g 4078 . . . . . . . . 9  |-  ( C  e.  Z  ->  C  e.  { A ,  B ,  C } )
2321, 22syl 17 . . . . . . . 8  |-  ( ph  ->  C  e.  { A ,  B ,  C }
)
2423adantr 472 . . . . . . 7  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  ->  C  e.  { A ,  B ,  C }
)
253, 24ffvelrnd 6038 . . . . . 6  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  -> 
( F `  C
)  e.  R )
2612, 20, 253jca 1210 . . . . 5  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  -> 
( ( F `  A )  e.  R  /\  ( F `  B
)  e.  R  /\  ( F `  C )  e.  R ) )
2726expcom 442 . . . 4  |-  ( F : { A ,  B ,  C } --> R  ->  ( ph  ->  ( ( F `  A
)  e.  R  /\  ( F `  B )  e.  R  /\  ( F `  C )  e.  R ) ) )
282, 27syl 17 . . 3  |-  ( F : { A ,  B ,  C } -1-1->
R  ->  ( ph  ->  ( ( F `  A )  e.  R  /\  ( F `  B
)  e.  R  /\  ( F `  C )  e.  R ) ) )
291, 28mpcom 36 . 2  |-  ( ph  ->  ( ( F `  A )  e.  R  /\  ( F `  B
)  e.  R  /\  ( F `  C )  e.  R ) )
30 f1dom3fv3dif.n . . 3  |-  ( ph  ->  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
316, 30, 1f1dom3fv3dif 6186 . 2  |-  ( ph  ->  ( ( F `  A )  =/=  ( F `  B )  /\  ( F `  A
)  =/=  ( F `
 C )  /\  ( F `  B )  =/=  ( F `  C ) ) )
32 neeq1 2705 . . . 4  |-  ( x  =  ( F `  A )  ->  (
x  =/=  y  <->  ( F `  A )  =/=  y
) )
33 neeq1 2705 . . . 4  |-  ( x  =  ( F `  A )  ->  (
x  =/=  z  <->  ( F `  A )  =/=  z
) )
3432, 333anbi12d 1366 . . 3  |-  ( x  =  ( F `  A )  ->  (
( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z
)  <->  ( ( F `
 A )  =/=  y  /\  ( F `
 A )  =/=  z  /\  y  =/=  z ) ) )
35 neeq2 2706 . . . 4  |-  ( y  =  ( F `  B )  ->  (
( F `  A
)  =/=  y  <->  ( F `  A )  =/=  ( F `  B )
) )
36 neeq1 2705 . . . 4  |-  ( y  =  ( F `  B )  ->  (
y  =/=  z  <->  ( F `  B )  =/=  z
) )
3735, 363anbi13d 1367 . . 3  |-  ( y  =  ( F `  B )  ->  (
( ( F `  A )  =/=  y  /\  ( F `  A
)  =/=  z  /\  y  =/=  z )  <->  ( ( F `  A )  =/=  ( F `  B
)  /\  ( F `  A )  =/=  z  /\  ( F `  B
)  =/=  z ) ) )
38 neeq2 2706 . . . 4  |-  ( z  =  ( F `  C )  ->  (
( F `  A
)  =/=  z  <->  ( F `  A )  =/=  ( F `  C )
) )
39 neeq2 2706 . . . 4  |-  ( z  =  ( F `  C )  ->  (
( F `  B
)  =/=  z  <->  ( F `  B )  =/=  ( F `  C )
) )
4038, 393anbi23d 1368 . . 3  |-  ( z  =  ( F `  C )  ->  (
( ( F `  A )  =/=  ( F `  B )  /\  ( F `  A
)  =/=  z  /\  ( F `  B )  =/=  z )  <->  ( ( F `  A )  =/=  ( F `  B
)  /\  ( F `  A )  =/=  ( F `  C )  /\  ( F `  B
)  =/=  ( F `
 C ) ) ) )
4134, 37, 40rspc3ev 3151 . 2  |-  ( ( ( ( F `  A )  e.  R  /\  ( F `  B
)  e.  R  /\  ( F `  C )  e.  R )  /\  ( ( F `  A )  =/=  ( F `  B )  /\  ( F `  A
)  =/=  ( F `
 C )  /\  ( F `  B )  =/=  ( F `  C ) ) )  ->  E. x  e.  R  E. y  e.  R  E. z  e.  R  ( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z
) )
4229, 31, 41syl2anc 673 1  |-  ( ph  ->  E. x  e.  R  E. y  e.  R  E. z  e.  R  ( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    \/ w3o 1006    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   {ctp 3963   -->wf 5585   -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-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fv 5597
This theorem is referenced by:  hashge3el3dif  12683
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