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Theorem iseqf1olemqf1o 10221
Description: Lemma for seq3f1o 10232. 
Q is a permutation of  ( M ... N
).  Q is formed from the constant portion of  J, followed by the single element  K (at position  K), followed by the rest of J (with the  K deleted and the elements before  K moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-Aug-2022.)
Hypotheses
Ref Expression
iseqf1olemqf.k  |-  ( ph  ->  K  e.  ( M ... N ) )
iseqf1olemqf.j  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemqf.q  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
Assertion
Ref Expression
iseqf1olemqf1o  |-  ( ph  ->  Q : ( M ... N ) -1-1-onto-> ( M ... N ) )
Distinct variable groups:    u, J    u, K    u, M    u, N    ph, u
Allowed substitution hint:    Q( u)

Proof of Theorem iseqf1olemqf1o
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqf1olemqf.k . . . 4  |-  ( ph  ->  K  e.  ( M ... N ) )
2 iseqf1olemqf.j . . . 4  |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )
3 iseqf1olemqf.q . . . 4  |-  Q  =  ( u  e.  ( M ... N ) 
|->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `  u
) ) )
41, 2, 3iseqf1olemqf 10219 . . 3  |-  ( ph  ->  Q : ( M ... N ) --> ( M ... N ) )
51ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  K  e.  ( M ... N ) )
62ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  J :
( M ... N
)
-1-1-onto-> ( M ... N ) )
7 simplrl 509 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  v  e.  ( M ... N ) )
8 simplrr 510 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  w  e.  ( M ... N ) )
9 simpr 109 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  ( Q `  v )  =  ( Q `  w ) )
105, 6, 3, 7, 8, 9iseqf1olemmo 10220 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( M ... N )  /\  w  e.  ( M ... N ) ) )  /\  ( Q `  v )  =  ( Q `  w ) )  ->  v  =  w )
1110ex 114 . . . 4  |-  ( (
ph  /\  ( v  e.  ( M ... N
)  /\  w  e.  ( M ... N ) ) )  ->  (
( Q `  v
)  =  ( Q `
 w )  -> 
v  =  w ) )
1211ralrimivva 2491 . . 3  |-  ( ph  ->  A. v  e.  ( M ... N ) A. w  e.  ( M ... N ) ( ( Q `  v )  =  ( Q `  w )  ->  v  =  w ) )
13 dff13 5637 . . 3  |-  ( Q : ( M ... N ) -1-1-> ( M ... N )  <->  ( Q : ( M ... N ) --> ( M ... N )  /\  A. v  e.  ( M ... N ) A. w  e.  ( M ... N ) ( ( Q `  v )  =  ( Q `  w )  ->  v  =  w ) ) )
144, 12, 13sylanbrc 413 . 2  |-  ( ph  ->  Q : ( M ... N ) -1-1-> ( M ... N ) )
15 elfzel1 9760 . . . . . 6  |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
161, 15syl 14 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
17 elfzel2 9759 . . . . . 6  |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
181, 17syl 14 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
1916, 18fzfigd 10159 . . . 4  |-  ( ph  ->  ( M ... N
)  e.  Fin )
20 enrefg 6626 . . . 4  |-  ( ( M ... N )  e.  Fin  ->  ( M ... N )  ~~  ( M ... N ) )
2119, 20syl 14 . . 3  |-  ( ph  ->  ( M ... N
)  ~~  ( M ... N ) )
22 f1finf1o 6803 . . 3  |-  ( ( ( M ... N
)  ~~  ( M ... N )  /\  ( M ... N )  e. 
Fin )  ->  ( Q : ( M ... N ) -1-1-> ( M ... N )  <->  Q :
( M ... N
)
-1-1-onto-> ( M ... N ) ) )
2321, 19, 22syl2anc 408 . 2  |-  ( ph  ->  ( Q : ( M ... N )
-1-1-> ( M ... N
)  <->  Q : ( M ... N ) -1-1-onto-> ( M ... N ) ) )
2414, 23mpbid 146 1  |-  ( ph  ->  Q : ( M ... N ) -1-1-onto-> ( M ... N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   A.wral 2393   ifcif 3444   class class class wbr 3899    |-> cmpt 3959   `'ccnv 4508   -->wf 5089   -1-1->wf1 5090   -1-1-onto->wf1o 5092   ` cfv 5093  (class class class)co 5742    ~~ cen 6600   Fincfn 6602   1c1 7589    - cmin 7901   ZZcz 9012   ...cfz 9745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-1o 6281  df-er 6397  df-en 6603  df-fin 6605  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8685  df-n0 8936  df-z 9013  df-uz 9283  df-fz 9746
This theorem is referenced by:  seq3f1olemqsumkj  10226  seq3f1olemqsumk  10227  seq3f1olemstep  10229
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