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Theorem nvnencycllem 24848
Description: Lemma for 3v3e3cycl1 24849 and 4cycl4v4e 24871. (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 455 . . . 4  |-  ( ( Fun  E  /\  F  e. Word  dom  E )  ->  Fun  E )
21adantr 463 . . 3  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  Fun  E )
3 wrdf 12541 . . . . . 6  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
43adantl 464 . . . . 5  |-  ( ( Fun  E  /\  F  e. Word  dom  E )  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
54adantr 463 . . . 4  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
6 lencl 12552 . . . . . . 7  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
7 simpl 455 . . . . . . . . . 10  |-  ( ( ( # `  F
)  e.  NN0  /\  ( X  e.  NN0  /\  X  <  ( # `  F ) ) )  ->  ( # `  F
)  e.  NN0 )
8 nn0ge0 10817 . . . . . . . . . . . . 13  |-  ( X  e.  NN0  ->  0  <_  X )
9 0red 9586 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  F
)  e.  NN0  /\  X  e.  NN0 )  -> 
0  e.  RR )
10 nn0re 10800 . . . . . . . . . . . . . . . . . . . 20  |-  ( X  e.  NN0  ->  X  e.  RR )
1110adantl 464 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  F
)  e.  NN0  /\  X  e.  NN0 )  ->  X  e.  RR )
12 nn0re 10800 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  RR )
1312adantr 463 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  F
)  e.  NN0  /\  X  e.  NN0 )  -> 
( # `  F )  e.  RR )
14 lelttr 9664 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  RR  /\  X  e.  RR  /\  ( # `
 F )  e.  RR )  ->  (
( 0  <_  X  /\  X  <  ( # `  F ) )  -> 
0  <  ( # `  F
) ) )
159, 11, 13, 14syl3anc 1226 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  F
)  e.  NN0  /\  X  e.  NN0 )  -> 
( ( 0  <_  X  /\  X  <  ( # `
 F ) )  ->  0  <  ( # `
 F ) ) )
16 ltne 9670 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  RR  /\  0  <  ( # `  F
) )  ->  ( # `
 F )  =/=  0 )
179, 15, 16syl6an 543 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  F
)  e.  NN0  /\  X  e.  NN0 )  -> 
( ( 0  <_  X  /\  X  <  ( # `
 F ) )  ->  ( # `  F
)  =/=  0 ) )
1817ex 432 . . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . . 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 427 . . . . . . . . . . 11  |-  ( ( X  e.  NN0  /\  X  <  ( # `  F
) )  ->  (
( # `  F )  e.  NN0  ->  ( # `  F )  =/=  0
) )
2423impcom 428 . . . . . . . . . 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 662 . . . . . . . . 9  |-  ( ( ( # `  F
)  e.  NN0  /\  ( X  e.  NN0  /\  X  <  ( # `  F ) ) )  ->  ( # `  F
)  e.  NN )
27 elfzo0 11840 . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . 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 429 . . . . . . . . . 10  |-  ( (
# `  F )  e.  NN  ->  ( ( X  e.  NN0  /\  X  <  ( # `  F
) )  ->  X  e.  ( 0..^ ( # `  F ) ) ) )
3231adantld 465 . . . . . . . . 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 432 . . . . . . 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 464 . . . . 5  |-  ( ( Fun  E  /\  F  e. Word  dom  E )  -> 
( ( X  e. 
NN0  /\  X  <  (
# `  F )
)  ->  X  e.  ( 0..^ ( # `  F
) ) ) )
3736imp 427 . . . 4  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  X  e.  ( 0..^ ( # `  F ) ) )
385, 37ffvelrnd 6008 . . 3  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  ( F `  X )  e.  dom  E )
39 fvelrn 6000 . . 3  |-  ( ( Fun  E  /\  ( F `  X )  e.  dom  E )  -> 
( E `  ( F `  X )
)  e.  ran  E
)
402, 38, 39syl2anc 659 . 2  |-  ( ( ( Fun  E  /\  F  e. Word  dom  E )  /\  ( X  e. 
NN0  /\  X  <  (
# `  F )
) )  ->  ( E `  ( F `  X ) )  e. 
ran  E )
41 eleq1 2526 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   {cpr 4018   class class class wbr 4439   dom cdm 4988   ran crn 4989   Fun wfun 5564   -->wf 5566   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481    < clt 9617    <_ cle 9618   NNcn 10531   NN0cn0 10791  ..^cfzo 11799   #chash 12390  Word cword 12521
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-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
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-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529
This theorem is referenced by:  3v3e3cycl1  24849  4cycl4v4e  24871  4cycl4dv  24872
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