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Theorem f1dom3el3dif 6151
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 5763 . . . 4  |-  ( F : { A ,  B ,  C } -1-1->
R  ->  F : { A ,  B ,  C } --> R )
3 simpr 459 . . . . . . 7  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  ->  F : { A ,  B ,  C } --> R )
4 eqidd 2455 . . . . . . . . . 10  |-  ( ph  ->  A  =  A )
543mix1d 1169 . . . . . . . . 9  |-  ( ph  ->  ( A  =  A  \/  A  =  B  \/  A  =  C ) )
6 f1dom3fv3dif.v . . . . . . . . . . 11  |-  ( ph  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
) )
76simp1d 1006 . . . . . . . . . 10  |-  ( ph  ->  A  e.  X )
8 eltpg 4058 . . . . . . . . . 10  |-  ( A  e.  X  ->  ( A  e.  { A ,  B ,  C }  <->  ( A  =  A  \/  A  =  B  \/  A  =  C )
) )
97, 8syl 16 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  { A ,  B ,  C }  <->  ( A  =  A  \/  A  =  B  \/  A  =  C ) ) )
105, 9mpbird 232 . . . . . . . 8  |-  ( ph  ->  A  e.  { A ,  B ,  C }
)
1110adantr 463 . . . . . . 7  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  ->  A  e.  { A ,  B ,  C }
)
123, 11ffvelrnd 6008 . . . . . 6  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  -> 
( F `  A
)  e.  R )
13 eqidd 2455 . . . . . . . . . 10  |-  ( ph  ->  B  =  B )
14133mix2d 1170 . . . . . . . . 9  |-  ( ph  ->  ( B  =  A  \/  B  =  B  \/  B  =  C ) )
156simp2d 1007 . . . . . . . . . 10  |-  ( ph  ->  B  e.  Y )
16 eltpg 4058 . . . . . . . . . 10  |-  ( B  e.  Y  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C )
) )
1715, 16syl 16 . . . . . . . . 9  |-  ( ph  ->  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C ) ) )
1814, 17mpbird 232 . . . . . . . 8  |-  ( ph  ->  B  e.  { A ,  B ,  C }
)
1918adantr 463 . . . . . . 7  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  ->  B  e.  { A ,  B ,  C }
)
203, 19ffvelrnd 6008 . . . . . 6  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  -> 
( F `  B
)  e.  R )
216simp3d 1008 . . . . . . . . 9  |-  ( ph  ->  C  e.  Z )
22 tpid3g 4131 . . . . . . . . 9  |-  ( C  e.  Z  ->  C  e.  { A ,  B ,  C } )
2321, 22syl 16 . . . . . . . 8  |-  ( ph  ->  C  e.  { A ,  B ,  C }
)
2423adantr 463 . . . . . . 7  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  ->  C  e.  { A ,  B ,  C }
)
253, 24ffvelrnd 6008 . . . . . 6  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  -> 
( F `  C
)  e.  R )
2612, 20, 253jca 1174 . . . . 5  |-  ( (
ph  /\  F : { A ,  B ,  C } --> R )  -> 
( ( F `  A )  e.  R  /\  ( F `  B
)  e.  R  /\  ( F `  C )  e.  R ) )
2726expcom 433 . . . 4  |-  ( F : { A ,  B ,  C } --> R  ->  ( ph  ->  ( ( F `  A
)  e.  R  /\  ( F `  B )  e.  R  /\  ( F `  C )  e.  R ) ) )
282, 27syl 16 . . 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 6150 . 2  |-  ( ph  ->  ( ( F `  A )  =/=  ( F `  B )  /\  ( F `  A
)  =/=  ( F `
 C )  /\  ( F `  B )  =/=  ( F `  C ) ) )
32 neeq1 2735 . . . 4  |-  ( x  =  ( F `  A )  ->  (
x  =/=  y  <->  ( F `  A )  =/=  y
) )
33 neeq1 2735 . . . 4  |-  ( x  =  ( F `  A )  ->  (
x  =/=  z  <->  ( F `  A )  =/=  z
) )
3432, 333anbi12d 1298 . . 3  |-  ( x  =  ( F `  A )  ->  (
( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z
)  <->  ( ( F `
 A )  =/=  y  /\  ( F `
 A )  =/=  z  /\  y  =/=  z ) ) )
35 neeq2 2737 . . . 4  |-  ( y  =  ( F `  B )  ->  (
( F `  A
)  =/=  y  <->  ( F `  A )  =/=  ( F `  B )
) )
36 neeq1 2735 . . . 4  |-  ( y  =  ( F `  B )  ->  (
y  =/=  z  <->  ( F `  B )  =/=  z
) )
3735, 363anbi13d 1299 . . 3  |-  ( y  =  ( F `  B )  ->  (
( ( F `  A )  =/=  y  /\  ( F `  A
)  =/=  z  /\  y  =/=  z )  <->  ( ( F `  A )  =/=  ( F `  B
)  /\  ( F `  A )  =/=  z  /\  ( F `  B
)  =/=  z ) ) )
38 neeq2 2737 . . . 4  |-  ( z  =  ( F `  C )  ->  (
( F `  A
)  =/=  z  <->  ( F `  A )  =/=  ( F `  C )
) )
39 neeq2 2737 . . . 4  |-  ( z  =  ( F `  C )  ->  (
( F `  B
)  =/=  z  <->  ( F `  B )  =/=  ( F `  C )
) )
4038, 393anbi23d 1300 . . 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 3220 . 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 659 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 184    /\ wa 367    \/ w3o 970    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   {ctp 4020   -->wf 5566   -1-1->wf1 5567   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fv 5578
This theorem is referenced by:  hashge3el3dif  12508
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