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Theorem s2f1o 11855
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 960 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  A  e.  S )
2 0z 10285 . . . . . 6  |-  0  e.  ZZ
31, 2jctil 524 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  ( 0  e.  ZZ  /\  A  e.  S ) )
4 simpl2 961 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  B  e.  S )
5 1z 10303 . . . . . 6  |-  1  e.  ZZ
64, 5jctil 524 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  ( 1  e.  ZZ  /\  B  e.  S ) )
73, 6jca 519 . . . 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 962 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  A  =/= 
B )
9 ax-1ne0 9051 . . . . . 6  |-  1  =/=  0
109necomi 2680 . . . . 5  |-  0  =/=  1
118, 10jctil 524 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  ( 0  =/=  1  /\  A  =/=  B ) )
12 f1oprg 5710 . . . 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 58 . . 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 2437 . . . . . 6  |-  ( E  =  <" A B ">  <->  <" A B ">  =  E )
15 s2prop 11853 . . . . . . . 8  |-  ( ( A  e.  S  /\  B  e.  S )  ->  <" A B ">  =  { <. 0 ,  A >. , 
<. 1 ,  B >. } )
16153adant3 977 . . . . . . 7  |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  ->  <" A B ">  =  { <. 0 ,  A >. ,  <. 1 ,  B >. } )
1716eqeq1d 2443 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  -> 
( <" A B ">  =  E  <->  { <. 0 ,  A >. ,  <. 1 ,  B >. }  =  E ) )
1814, 17syl5bb 249 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  -> 
( E  =  <" A B ">  <->  { <. 0 ,  A >. , 
<. 1 ,  B >. }  =  E ) )
1918biimpa 471 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  { <. 0 ,  A >. , 
<. 1 ,  B >. }  =  E )
20 eqidd 2436 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  { 0 ,  1 }  =  { 0 ,  1 } )
21 eqidd 2436 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  { A ,  B }  =  { A ,  B }
)
2219, 20, 21f1oeq123d 5663 . . 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 202 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B
)  /\  E  =  <" A B "> )  ->  E : { 0 ,  1 } -1-1-onto-> { A ,  B } )
2423ex 424 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   {cpr 3807   <.cop 3809   -1-1-onto->wf1o 5445   0cc0 8982   1c1 8983   ZZcz 10274   <"cs2 11797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-hash 11611  df-word 11715  df-concat 11716  df-s1 11717  df-s2 11804
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