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Theorem frecfzennn 9367
Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 9349 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
Hypothesis
Ref Expression
frecfzennn.1  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
Assertion
Ref Expression
frecfzennn  |-  ( N  e.  NN0  ->  ( 1 ... N )  ~~  ( `' G `  N ) )

Proof of Theorem frecfzennn
Dummy variables  m  n  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5548 . . 3  |-  ( n  =  0  ->  (
1 ... n )  =  ( 1 ... 0
) )
2 fveq2 5206 . . 3  |-  ( n  =  0  ->  ( `' G `  n )  =  ( `' G `  0 ) )
31, 2breq12d 3805 . 2  |-  ( n  =  0  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... 0 ) 
~~  ( `' G `  0 ) ) )
4 oveq2 5548 . . 3  |-  ( n  =  m  ->  (
1 ... n )  =  ( 1 ... m
) )
5 fveq2 5206 . . 3  |-  ( n  =  m  ->  ( `' G `  n )  =  ( `' G `  m ) )
64, 5breq12d 3805 . 2  |-  ( n  =  m  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... m ) 
~~  ( `' G `  m ) ) )
7 oveq2 5548 . . 3  |-  ( n  =  ( m  + 
1 )  ->  (
1 ... n )  =  ( 1 ... (
m  +  1 ) ) )
8 fveq2 5206 . . 3  |-  ( n  =  ( m  + 
1 )  ->  ( `' G `  n )  =  ( `' G `  ( m  +  1 ) ) )
97, 8breq12d 3805 . 2  |-  ( n  =  ( m  + 
1 )  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... ( m  +  1 ) ) 
~~  ( `' G `  ( m  +  1 ) ) ) )
10 oveq2 5548 . . 3  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
11 fveq2 5206 . . 3  |-  ( n  =  N  ->  ( `' G `  n )  =  ( `' G `  N ) )
1210, 11breq12d 3805 . 2  |-  ( n  =  N  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... N ) 
~~  ( `' G `  N ) ) )
13 0ex 3912 . . . 4  |-  (/)  e.  _V
1413enref 6276 . . 3  |-  (/)  ~~  (/)
15 fz10 9012 . . 3  |-  ( 1 ... 0 )  =  (/)
16 0zd 8314 . . . . . . 7  |-  ( T. 
->  0  e.  ZZ )
17 frecfzennn.1 . . . . . . 7  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
1816, 17frec2uzf1od 9356 . . . . . 6  |-  ( T. 
->  G : om -1-1-onto-> ( ZZ>= `  0 )
)
1918trud 1268 . . . . 5  |-  G : om
-1-1-onto-> ( ZZ>= `  0 )
20 peano1 4345 . . . . 5  |-  (/)  e.  om
2119, 20pm3.2i 261 . . . 4  |-  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )
2216, 17frec2uz0d 9349 . . . . 5  |-  ( T. 
->  ( G `  (/) )  =  0 )
2322trud 1268 . . . 4  |-  ( G `
 (/) )  =  0
24 f1ocnvfv 5447 . . . 4  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )  ->  ( ( G `  (/) )  =  0  -> 
( `' G ` 
0 )  =  (/) ) )
2521, 23, 24mp2 16 . . 3  |-  ( `' G `  0 )  =  (/)
2614, 15, 253brtr4i 3820 . 2  |-  ( 1 ... 0 )  ~~  ( `' G `  0 )
27 simpr 107 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... m )  ~~  ( `' G `  m ) )
28 peano2nn0 8279 . . . . . . 7  |-  ( m  e.  NN0  ->  ( m  +  1 )  e. 
NN0 )
29 zex 8311 . . . . . . . . . . . . . . 15  |-  ZZ  e.  _V
3029mptex 5415 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  |->  ( x  +  1 ) )  e.  _V
31 vex 2577 . . . . . . . . . . . . . 14  |-  z  e. 
_V
3230, 31fvex 5223 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  |->  ( x  +  1 ) ) `  z )  e.  _V
3332ax-gen 1354 . . . . . . . . . . . 12  |-  A. z
( ( x  e.  ZZ  |->  ( x  + 
1 ) ) `  z )  e.  _V
34 0z 8313 . . . . . . . . . . . 12  |-  0  e.  ZZ
35 frecfnom 6017 . . . . . . . . . . . 12  |-  ( ( A. z ( ( x  e.  ZZ  |->  ( x  +  1 ) ) `  z )  e.  _V  /\  0  e.  ZZ )  -> frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  Fn  om )
3633, 34, 35mp2an 410 . . . . . . . . . . 11  |- frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  Fn  om
3717fneq1i 5021 . . . . . . . . . . 11  |-  ( G  Fn  om  <-> frec ( (
x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  Fn  om )
3836, 37mpbir 138 . . . . . . . . . 10  |-  G  Fn  om
39 omex 4344 . . . . . . . . . 10  |-  om  e.  _V
40 fnex 5411 . . . . . . . . . 10  |-  ( ( G  Fn  om  /\  om  e.  _V )  ->  G  e.  _V )
4138, 39, 40mp2an 410 . . . . . . . . 9  |-  G  e. 
_V
4241cnvex 4884 . . . . . . . 8  |-  `' G  e.  _V
43 vex 2577 . . . . . . . 8  |-  m  e. 
_V
4442, 43fvex 5223 . . . . . . 7  |-  ( `' G `  m )  e.  _V
45 en2sn 6321 . . . . . . 7  |-  ( ( ( m  +  1 )  e.  NN0  /\  ( `' G `  m )  e.  _V )  ->  { ( m  + 
1 ) }  ~~  { ( `' G `  m ) } )
4628, 44, 45sylancl 398 . . . . . 6  |-  ( m  e.  NN0  ->  { ( m  +  1 ) }  ~~  { ( `' G `  m ) } )
4746adantr 265 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  { (
m  +  1 ) }  ~~  { ( `' G `  m ) } )
48 fzp1disj 9044 . . . . . 6  |-  ( ( 1 ... m )  i^i  { ( m  +  1 ) } )  =  (/)
4948a1i 9 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( (
1 ... m )  i^i 
{ ( m  + 
1 ) } )  =  (/) )
50 f1ocnvdm 5449 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  m  e.  ( ZZ>=
`  0 ) )  ->  ( `' G `  m )  e.  om )
5119, 50mpan 408 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  0
)  ->  ( `' G `  m )  e.  om )
52 nn0uz 8603 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
5351, 52eleq2s 2148 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( `' G `  m )  e.  om )
54 nnord 4362 . . . . . . . 8  |-  ( ( `' G `  m )  e.  om  ->  Ord  ( `' G `  m ) )
55 ordirr 4295 . . . . . . . 8  |-  ( Ord  ( `' G `  m )  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
5653, 54, 553syl 17 . . . . . . 7  |-  ( m  e.  NN0  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
5756adantr 265 . . . . . 6  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
58 disjsn 3460 . . . . . 6  |-  ( ( ( `' G `  m )  i^i  {
( `' G `  m ) } )  =  (/)  <->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
5957, 58sylibr 141 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( ( `' G `  m )  i^i  { ( `' G `  m ) } )  =  (/) )
60 unen 6324 . . . . 5  |-  ( ( ( ( 1 ... m )  ~~  ( `' G `  m )  /\  { ( m  +  1 ) } 
~~  { ( `' G `  m ) } )  /\  (
( ( 1 ... m )  i^i  {
( m  +  1 ) } )  =  (/)  /\  ( ( `' G `  m )  i^i  { ( `' G `  m ) } )  =  (/) ) )  ->  (
( 1 ... m
)  u.  { ( m  +  1 ) } )  ~~  (
( `' G `  m )  u.  {
( `' G `  m ) } ) )
6127, 47, 49, 59, 60syl22anc 1147 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( (
1 ... m )  u. 
{ ( m  + 
1 ) } ) 
~~  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
62 1z 8328 . . . . . 6  |-  1  e.  ZZ
63 1m1e0 8059 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
6463fveq2i 5209 . . . . . . . . 9  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
6552, 64eqtr4i 2079 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
6665eleq2i 2120 . . . . . . 7  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  ( 1  -  1 ) ) )
6766biimpi 117 . . . . . 6  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )
68 fzsuc2 9043 . . . . . 6  |-  ( ( 1  e.  ZZ  /\  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )  -> 
( 1 ... (
m  +  1 ) )  =  ( ( 1 ... m )  u.  { ( m  +  1 ) } ) )
6962, 67, 68sylancr 399 . . . . 5  |-  ( m  e.  NN0  ->  ( 1 ... ( m  + 
1 ) )  =  ( ( 1 ... m )  u.  {
( m  +  1 ) } ) )
7069adantr 265 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... ( m  + 
1 ) )  =  ( ( 1 ... m )  u.  {
( m  +  1 ) } ) )
71 peano2 4346 . . . . . . . . 9  |-  ( ( `' G `  m )  e.  om  ->  suc  ( `' G `  m )  e.  om )
7253, 71syl 14 . . . . . . . 8  |-  ( m  e.  NN0  ->  suc  ( `' G `  m )  e.  om )
7372, 19jctil 299 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  suc  ( `' G `  m )  e.  om ) )
74 0zd 8314 . . . . . . . . . 10  |-  ( ( `' G `  m )  e.  om  ->  0  e.  ZZ )
75 id 19 . . . . . . . . . 10  |-  ( ( `' G `  m )  e.  om  ->  ( `' G `  m )  e.  om )
7674, 17, 75frec2uzsucd 9351 . . . . . . . . 9  |-  ( ( `' G `  m )  e.  om  ->  ( G `  suc  ( `' G `  m ) )  =  ( ( G `  ( `' G `  m ) )  +  1 ) )
7753, 76syl 14 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( G `
 suc  ( `' G `  m )
)  =  ( ( G `  ( `' G `  m ) )  +  1 ) )
7852eleq2i 2120 . . . . . . . . . . 11  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  0 ) )
7978biimpi 117 . . . . . . . . . 10  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  0 )
)
80 f1ocnvfv2 5446 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  m  e.  ( ZZ>=
`  0 ) )  ->  ( G `  ( `' G `  m ) )  =  m )
8119, 79, 80sylancr 399 . . . . . . . . 9  |-  ( m  e.  NN0  ->  ( G `
 ( `' G `  m ) )  =  m )
8281oveq1d 5555 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( ( G `  ( `' G `  m ) )  +  1 )  =  ( m  + 
1 ) )
8377, 82eqtrd 2088 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G `
 suc  ( `' G `  m )
)  =  ( m  +  1 ) )
84 f1ocnvfv 5447 . . . . . . 7  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  suc  ( `' G `  m )  e.  om )  ->  ( ( G `
 suc  ( `' G `  m )
)  =  ( m  +  1 )  -> 
( `' G `  ( m  +  1
) )  =  suc  ( `' G `  m ) ) )
8573, 83, 84sylc 60 . . . . . 6  |-  ( m  e.  NN0  ->  ( `' G `  ( m  +  1 ) )  =  suc  ( `' G `  m ) )
8685adantr 265 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  suc  ( `' G `  m ) )
87 df-suc 4136 . . . . 5  |-  suc  ( `' G `  m )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } )
8886, 87syl6eq 2104 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
8961, 70, 883brtr4d 3822 . . 3  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... ( m  + 
1 ) )  ~~  ( `' G `  ( m  +  1 ) ) )
9089ex 112 . 2  |-  ( m  e.  NN0  ->  ( ( 1 ... m ) 
~~  ( `' G `  m )  ->  (
1 ... ( m  + 
1 ) )  ~~  ( `' G `  ( m  +  1 ) ) ) )
913, 6, 9, 12, 26, 90nn0ind 8411 1  |-  ( N  e.  NN0  ->  ( 1 ... N )  ~~  ( `' G `  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101   A.wal 1257    = wceq 1259   T. wtru 1260    e. wcel 1409   _Vcvv 2574    u. cun 2943    i^i cin 2944   (/)c0 3252   {csn 3403   class class class wbr 3792    |-> cmpt 3846   Ord word 4127   suc csuc 4130   omcom 4341   `'ccnv 4372    Fn wfn 4925   -1-1-onto->wf1o 4929   ` cfv 4930  (class class class)co 5540  freccfrec 6008    ~~ cen 6250   0cc0 6947   1c1 6948    + caddc 6950    - cmin 7245   NN0cn0 8239   ZZcz 8302   ZZ>=cuz 8569   ...cfz 8976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-en 6253  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-fz 8977
This theorem is referenced by:  frecfzen2  9368
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