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Theorem s2f1o 27299
Description: A length 2 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
Assertion
Ref Expression
s2f1o  |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  -> 
( E  =  <" A B ">  ->  E : { 0 ,  1 } -1-1-onto-> { A ,  B } ) )

Proof of Theorem s2f1o
StepHypRef Expression
1 simpl1 958 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  A  e.  S )
2 0z 10082 . . . . . 6  |-  0  e.  ZZ
31, 2jctil 523 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  ( 0  e.  ZZ  /\  A  e.  S ) )
4 simpl2 959 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  B  e.  S )
5 1z 10100 . . . . . 6  |-  1  e.  ZZ
64, 5jctil 523 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  ( 1  e.  ZZ  /\  B  e.  S ) )
73, 6jca 518 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  ( ( 0  e.  ZZ  /\  A  e.  S )  /\  ( 1  e.  ZZ  /\  B  e.  S ) ) )
8 simpl3 960 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  A  =/= 
B )
9 ax-1ne0 8851 . . . . . 6  |-  1  =/=  0
109necomi 2561 . . . . 5  |-  0  =/=  1
118, 10jctil 523 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  ( 0  =/=  1  /\  A  =/=  B ) )
12 f1oprg 27240 . . . 4  |-  ( ( ( 0  e.  ZZ  /\  A  e.  S )  /\  ( 1  e.  ZZ  /\  B  e.  S ) )  -> 
( ( 0  =/=  1  /\  A  =/= 
B )  ->  { <. 0 ,  A >. , 
<. 1 ,  B >. } : { 0 ,  1 } -1-1-onto-> { A ,  B } ) )
137, 11, 12sylc 56 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  { <. 0 ,  A >. , 
<. 1 ,  B >. } : { 0 ,  1 } -1-1-onto-> { A ,  B } )
14 eqcom 2318 . . . . . 6  |-  ( E  =  <" A B ">  <->  <" A B ">  =  E )
15 s2prop 27297 . . . . . . . 8  |-  ( ( A  e.  S  /\  B  e.  S )  ->  <" A B ">  =  { <. 0 ,  A >. , 
<. 1 ,  B >. } )
16153adant3 975 . . . . . . 7  |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  ->  <" A B ">  =  { <. 0 ,  A >. ,  <. 1 ,  B >. } )
1716eqeq1d 2324 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  -> 
( <" A B ">  =  E  <->  { <. 0 ,  A >. ,  <. 1 ,  B >. }  =  E ) )
1814, 17syl5bb 248 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  -> 
( E  =  <" A B ">  <->  { <. 0 ,  A >. , 
<. 1 ,  B >. }  =  E ) )
1918biimpa 470 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  { <. 0 ,  A >. , 
<. 1 ,  B >. }  =  E )
20 eqidd 2317 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  { 0 ,  1 }  =  { 0 ,  1 } )
21 eqidd 2317 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  { A ,  B }  =  { A ,  B }
)
2219, 20, 21f1oeq123d 5507 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  ( {
<. 0 ,  A >. ,  <. 1 ,  B >. } : { 0 ,  1 } -1-1-onto-> { A ,  B } 
<->  E : { 0 ,  1 } -1-1-onto-> { A ,  B } ) )
2313, 22mpbid 201 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  E : { 0 ,  1 } -1-1-onto-> { A ,  B } )
2423ex 423 1  |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  -> 
( E  =  <" A B ">  ->  E : { 0 ,  1 } -1-1-onto-> { A ,  B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   {cpr 3675   <.cop 3677   -1-1-onto->wf1o 5291   0cc0 8782   1c1 8783   ZZcz 10071   <"cs2 11538
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-fzo 10918  df-hash 11385  df-word 11456  df-concat 11457  df-s1 11458  df-s2 11545
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