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Theorem nvnencycllem 24319
Description: Lemma for 3v3e3cycl1 24320 and 4cycl4v4e 24342. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
Assertion
Ref Expression
nvnencycllem  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  (
( E `  ( F `  X )
)  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )

Proof of Theorem nvnencycllem
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( Fun  E  /\  F  e. Word  dom  E )  ->  Fun  E )
21adantr 465 . . 3  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  Fun  E )
3 wrdf 12515 . . . . . 6  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
43adantl 466 . . . . 5  |-  ( ( Fun  E  /\  F  e. Word  dom  E )  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
54adantr 465 . . . 4  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
6 lencl 12524 . . . . . . 7  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
7 simpl 457 . . . . . . . . . 10  |-  ( ( ( # `  F
)  e.  NN0  /\  ( X  e.  NN0  /\  X  <  ( # `  F ) ) )  ->  ( # `  F
)  e.  NN0 )
8 nn0ge0 10817 . . . . . . . . . . . . 13  |-  ( X  e.  NN0  ->  0  <_  X )
9 0red 9593 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  F
)  e.  NN0  /\  X  e.  NN0 )  -> 
0  e.  RR )
10 nn0re 10800 . . . . . . . . . . . . . . . . . . . 20  |-  ( X  e.  NN0  ->  X  e.  RR )
1110adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  F
)  e.  NN0  /\  X  e.  NN0 )  ->  X  e.  RR )
12 nn0re 10800 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  RR )
1312adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  F
)  e.  NN0  /\  X  e.  NN0 )  -> 
( # `  F )  e.  RR )
14 lelttr 9671 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  RR  /\  X  e.  RR  /\  ( # `
 F )  e.  RR )  ->  (
( 0  <_  X  /\  X  <  ( # `  F ) )  -> 
0  <  ( # `  F
) ) )
159, 11, 13, 14syl3anc 1228 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  F
)  e.  NN0  /\  X  e.  NN0 )  -> 
( ( 0  <_  X  /\  X  <  ( # `
 F ) )  ->  0  <  ( # `
 F ) ) )
16 ltne 9677 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  RR  /\  0  <  ( # `  F
) )  ->  ( # `
 F )  =/=  0 )
179, 15, 16syl6an 545 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  F
)  e.  NN0  /\  X  e.  NN0 )  -> 
( ( 0  <_  X  /\  X  <  ( # `
 F ) )  ->  ( # `  F
)  =/=  0 ) )
1817ex 434 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  NN0  ->  ( X  e.  NN0  ->  ( (
0  <_  X  /\  X  <  ( # `  F
) )  ->  ( # `
 F )  =/=  0 ) ) )
1918com13 80 . . . . . . . . . . . . . . 15  |-  ( ( 0  <_  X  /\  X  <  ( # `  F
) )  ->  ( X  e.  NN0  ->  (
( # `  F )  e.  NN0  ->  ( # `  F )  =/=  0
) ) )
2019ex 434 . . . . . . . . . . . . . 14  |-  ( 0  <_  X  ->  ( X  <  ( # `  F
)  ->  ( X  e.  NN0  ->  ( ( # `
 F )  e. 
NN0  ->  ( # `  F
)  =/=  0 ) ) ) )
2120com23 78 . . . . . . . . . . . . 13  |-  ( 0  <_  X  ->  ( X  e.  NN0  ->  ( X  <  ( # `  F
)  ->  ( ( # `
 F )  e. 
NN0  ->  ( # `  F
)  =/=  0 ) ) ) )
228, 21mpcom 36 . . . . . . . . . . . 12  |-  ( X  e.  NN0  ->  ( X  <  ( # `  F
)  ->  ( ( # `
 F )  e. 
NN0  ->  ( # `  F
)  =/=  0 ) ) )
2322imp 429 . . . . . . . . . . 11  |-  ( ( X  e.  NN0  /\  X  <  ( # `  F
) )  ->  (
( # `  F )  e.  NN0  ->  ( # `  F )  =/=  0
) )
2423impcom 430 . . . . . . . . . 10  |-  ( ( ( # `  F
)  e.  NN0  /\  ( X  e.  NN0  /\  X  <  ( # `  F ) ) )  ->  ( # `  F
)  =/=  0 )
25 elnnne0 10805 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN  <->  ( ( # `  F )  e.  NN0  /\  ( # `  F
)  =/=  0 ) )
267, 24, 25sylanbrc 664 . . . . . . . . 9  |-  ( ( ( # `  F
)  e.  NN0  /\  ( X  e.  NN0  /\  X  <  ( # `  F ) ) )  ->  ( # `  F
)  e.  NN )
27 elfzo0 11827 . . . . . . . . . . . . . 14  |-  ( X  e.  ( 0..^ (
# `  F )
)  <->  ( X  e. 
NN0  /\  ( # `  F
)  e.  NN  /\  X  <  ( # `  F
) ) )
2827biimpri 206 . . . . . . . . . . . . 13  |-  ( ( X  e.  NN0  /\  ( # `  F )  e.  NN  /\  X  <  ( # `  F
) )  ->  X  e.  ( 0..^ ( # `  F ) ) )
29283exp 1195 . . . . . . . . . . . 12  |-  ( X  e.  NN0  ->  ( (
# `  F )  e.  NN  ->  ( X  <  ( # `  F
)  ->  X  e.  ( 0..^ ( # `  F
) ) ) ) )
3029com12 31 . . . . . . . . . . 11  |-  ( (
# `  F )  e.  NN  ->  ( X  e.  NN0  ->  ( X  <  ( # `  F
)  ->  X  e.  ( 0..^ ( # `  F
) ) ) ) )
3130impd 431 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN  ->  ( ( X  e.  NN0  /\  X  <  ( # `  F
) )  ->  X  e.  ( 0..^ ( # `  F ) ) ) )
3231adantld 467 . . . . . . . . 9  |-  ( (
# `  F )  e.  NN  ->  ( (
( # `  F )  e.  NN0  /\  ( X  e.  NN0  /\  X  <  ( # `  F
) ) )  ->  X  e.  ( 0..^ ( # `  F
) ) ) )
3326, 32mpcom 36 . . . . . . . 8  |-  ( ( ( # `  F
)  e.  NN0  /\  ( X  e.  NN0  /\  X  <  ( # `  F ) ) )  ->  X  e.  ( 0..^ ( # `  F
) ) )
3433ex 434 . . . . . . 7  |-  ( (
# `  F )  e.  NN0  ->  ( ( X  e.  NN0  /\  X  <  ( # `  F
) )  ->  X  e.  ( 0..^ ( # `  F ) ) ) )
356, 34syl 16 . . . . . 6  |-  ( F  e. Word  dom  E  ->  ( ( X  e.  NN0  /\  X  <  ( # `  F ) )  ->  X  e.  ( 0..^ ( # `  F
) ) ) )
3635adantl 466 . . . . 5  |-  ( ( Fun  E  /\  F  e. Word  dom  E )  -> 
( ( X  e. 
NN0  /\  X  <  (
# `  F )
)  ->  X  e.  ( 0..^ ( # `  F
) ) ) )
3736imp 429 . . . 4  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  X  e.  ( 0..^ ( # `  F ) ) )
385, 37ffvelrnd 6020 . . 3  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  ( F `  X )  e.  dom  E )
39 fvelrn 6015 . . 3  |-  ( ( Fun  E  /\  ( F `  X )  e.  dom  E )  -> 
( E `  ( F `  X )
)  e.  ran  E
)
402, 38, 39syl2anc 661 . 2  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  ( E `  ( F `  X ) )  e. 
ran  E )
41 eleq1 2539 . 2  |-  ( ( E `  ( F `
 X ) )  =  { A ,  B }  ->  ( ( E `  ( F `
 X ) )  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
4240, 41syl5ibcom 220 1  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  (
( E `  ( F `  X )
)  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {cpr 4029   class class class wbr 4447   dom cdm 4999   ran crn 5000   Fun wfun 5580   -->wf 5582   ` cfv 5586  (class class class)co 6282   RRcr 9487   0cc0 9488    < clt 9624    <_ cle 9625   NNcn 10532   NN0cn0 10791  ..^cfzo 11788   #chash 12369  Word cword 12496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504
This theorem is referenced by:  3v3e3cycl1  24320  4cycl4v4e  24342  4cycl4dv  24343
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