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Theorem frecfzennn 9798
Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 9771 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 5642 . . 3  |-  ( n  =  0  ->  (
1 ... n )  =  ( 1 ... 0
) )
2 fveq2 5289 . . 3  |-  ( n  =  0  ->  ( `' G `  n )  =  ( `' G `  0 ) )
31, 2breq12d 3850 . 2  |-  ( n  =  0  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... 0 ) 
~~  ( `' G `  0 ) ) )
4 oveq2 5642 . . 3  |-  ( n  =  m  ->  (
1 ... n )  =  ( 1 ... m
) )
5 fveq2 5289 . . 3  |-  ( n  =  m  ->  ( `' G `  n )  =  ( `' G `  m ) )
64, 5breq12d 3850 . 2  |-  ( n  =  m  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... m ) 
~~  ( `' G `  m ) ) )
7 oveq2 5642 . . 3  |-  ( n  =  ( m  + 
1 )  ->  (
1 ... n )  =  ( 1 ... (
m  +  1 ) ) )
8 fveq2 5289 . . 3  |-  ( n  =  ( m  + 
1 )  ->  ( `' G `  n )  =  ( `' G `  ( m  +  1 ) ) )
97, 8breq12d 3850 . 2  |-  ( n  =  ( m  + 
1 )  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... ( m  +  1 ) ) 
~~  ( `' G `  ( m  +  1 ) ) ) )
10 oveq2 5642 . . 3  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
11 fveq2 5289 . . 3  |-  ( n  =  N  ->  ( `' G `  n )  =  ( `' G `  N ) )
1210, 11breq12d 3850 . 2  |-  ( n  =  N  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... N ) 
~~  ( `' G `  N ) ) )
13 0ex 3958 . . . 4  |-  (/)  e.  _V
1413enref 6462 . . 3  |-  (/)  ~~  (/)
15 fz10 9429 . . 3  |-  ( 1 ... 0 )  =  (/)
16 0zd 8732 . . . . . . 7  |-  ( T. 
->  0  e.  ZZ )
17 frecfzennn.1 . . . . . . 7  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
1816, 17frec2uzf1od 9778 . . . . . 6  |-  ( T. 
->  G : om -1-1-onto-> ( ZZ>= `  0 )
)
1918mptru 1298 . . . . 5  |-  G : om
-1-1-onto-> ( ZZ>= `  0 )
20 peano1 4399 . . . . 5  |-  (/)  e.  om
2119, 20pm3.2i 266 . . . 4  |-  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )
2216, 17frec2uz0d 9771 . . . . 5  |-  ( T. 
->  ( G `  (/) )  =  0 )
2322mptru 1298 . . . 4  |-  ( G `
 (/) )  =  0
24 f1ocnvfv 5540 . . . 4  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )  ->  ( ( G `  (/) )  =  0  -> 
( `' G ` 
0 )  =  (/) ) )
2521, 23, 24mp2 16 . . 3  |-  ( `' G `  0 )  =  (/)
2614, 15, 253brtr4i 3865 . 2  |-  ( 1 ... 0 )  ~~  ( `' G `  0 )
27 simpr 108 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... m )  ~~  ( `' G `  m ) )
28 peano2nn0 8683 . . . . . . 7  |-  ( m  e.  NN0  ->  ( m  +  1 )  e. 
NN0 )
29 zex 8729 . . . . . . . . . . . . . . 15  |-  ZZ  e.  _V
3029mptex 5505 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  |->  ( x  +  1 ) )  e.  _V
31 vex 2622 . . . . . . . . . . . . . 14  |-  z  e. 
_V
3230, 31fvex 5309 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  |->  ( x  +  1 ) ) `  z )  e.  _V
3332ax-gen 1383 . . . . . . . . . . . 12  |-  A. z
( ( x  e.  ZZ  |->  ( x  + 
1 ) ) `  z )  e.  _V
34 0z 8731 . . . . . . . . . . . 12  |-  0  e.  ZZ
35 frecfnom 6148 . . . . . . . . . . . 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 417 . . . . . . . . . . 11  |- frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  Fn  om
3717fneq1i 5094 . . . . . . . . . . 11  |-  ( G  Fn  om  <-> frec ( (
x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  Fn  om )
3836, 37mpbir 144 . . . . . . . . . 10  |-  G  Fn  om
39 omex 4398 . . . . . . . . . 10  |-  om  e.  _V
40 fnex 5501 . . . . . . . . . 10  |-  ( ( G  Fn  om  /\  om  e.  _V )  ->  G  e.  _V )
4138, 39, 40mp2an 417 . . . . . . . . 9  |-  G  e. 
_V
4241cnvex 4956 . . . . . . . 8  |-  `' G  e.  _V
43 vex 2622 . . . . . . . 8  |-  m  e. 
_V
4442, 43fvex 5309 . . . . . . 7  |-  ( `' G `  m )  e.  _V
45 en2sn 6510 . . . . . . 7  |-  ( ( ( m  +  1 )  e.  NN0  /\  ( `' G `  m )  e.  _V )  ->  { ( m  + 
1 ) }  ~~  { ( `' G `  m ) } )
4628, 44, 45sylancl 404 . . . . . 6  |-  ( m  e.  NN0  ->  { ( m  +  1 ) }  ~~  { ( `' G `  m ) } )
4746adantr 270 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  { (
m  +  1 ) }  ~~  { ( `' G `  m ) } )
48 fzp1disj 9461 . . . . . 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 5542 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  m  e.  ( ZZ>=
`  0 ) )  ->  ( `' G `  m )  e.  om )
5119, 50mpan 415 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  0
)  ->  ( `' G `  m )  e.  om )
52 nn0uz 9022 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
5351, 52eleq2s 2182 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( `' G `  m )  e.  om )
54 nnord 4416 . . . . . . . 8  |-  ( ( `' G `  m )  e.  om  ->  Ord  ( `' G `  m ) )
55 ordirr 4348 . . . . . . . 8  |-  ( Ord  ( `' G `  m )  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
5653, 54, 553syl 17 . . . . . . 7  |-  ( m  e.  NN0  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
5756adantr 270 . . . . . 6  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
58 disjsn 3499 . . . . . 6  |-  ( ( ( `' G `  m )  i^i  {
( `' G `  m ) } )  =  (/)  <->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
5957, 58sylibr 132 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( ( `' G `  m )  i^i  { ( `' G `  m ) } )  =  (/) )
60 unen 6513 . . . . 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 1175 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( (
1 ... m )  u. 
{ ( m  + 
1 ) } ) 
~~  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
62 1z 8746 . . . . . 6  |-  1  e.  ZZ
63 1m1e0 8462 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
6463fveq2i 5292 . . . . . . . . 9  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
6552, 64eqtr4i 2111 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
6665eleq2i 2154 . . . . . . 7  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  ( 1  -  1 ) ) )
6766biimpi 118 . . . . . 6  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )
68 fzsuc2 9460 . . . . . 6  |-  ( ( 1  e.  ZZ  /\  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )  -> 
( 1 ... (
m  +  1 ) )  =  ( ( 1 ... m )  u.  { ( m  +  1 ) } ) )
6962, 67, 68sylancr 405 . . . . 5  |-  ( m  e.  NN0  ->  ( 1 ... ( m  + 
1 ) )  =  ( ( 1 ... m )  u.  {
( m  +  1 ) } ) )
7069adantr 270 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... ( m  + 
1 ) )  =  ( ( 1 ... m )  u.  {
( m  +  1 ) } ) )
71 peano2 4400 . . . . . . . . 9  |-  ( ( `' G `  m )  e.  om  ->  suc  ( `' G `  m )  e.  om )
7253, 71syl 14 . . . . . . . 8  |-  ( m  e.  NN0  ->  suc  ( `' G `  m )  e.  om )
7372, 19jctil 305 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  suc  ( `' G `  m )  e.  om ) )
74 0zd 8732 . . . . . . . . . 10  |-  ( ( `' G `  m )  e.  om  ->  0  e.  ZZ )
75 id 19 . . . . . . . . . 10  |-  ( ( `' G `  m )  e.  om  ->  ( `' G `  m )  e.  om )
7674, 17, 75frec2uzsucd 9773 . . . . . . . . 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 2154 . . . . . . . . . . 11  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  0 ) )
7978biimpi 118 . . . . . . . . . 10  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  0 )
)
80 f1ocnvfv2 5539 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  m  e.  ( ZZ>=
`  0 ) )  ->  ( G `  ( `' G `  m ) )  =  m )
8119, 79, 80sylancr 405 . . . . . . . . 9  |-  ( m  e.  NN0  ->  ( G `
 ( `' G `  m ) )  =  m )
8281oveq1d 5649 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( ( G `  ( `' G `  m ) )  +  1 )  =  ( m  + 
1 ) )
8377, 82eqtrd 2120 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G `
 suc  ( `' G `  m )
)  =  ( m  +  1 ) )
84 f1ocnvfv 5540 . . . . . . 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 61 . . . . . 6  |-  ( m  e.  NN0  ->  ( `' G `  ( m  +  1 ) )  =  suc  ( `' G `  m ) )
8685adantr 270 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  suc  ( `' G `  m ) )
87 df-suc 4189 . . . . 5  |-  suc  ( `' G `  m )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } )
8886, 87syl6eq 2136 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
8961, 70, 883brtr4d 3867 . . 3  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... ( m  + 
1 ) )  ~~  ( `' G `  ( m  +  1 ) ) )
9089ex 113 . 2  |-  ( m  e.  NN0  ->  ( ( 1 ... m ) 
~~  ( `' G `  m )  ->  (
1 ... ( m  + 
1 ) )  ~~  ( `' G `  ( m  +  1 ) ) ) )
913, 6, 9, 12, 26, 90nn0ind 8830 1  |-  ( N  e.  NN0  ->  ( 1 ... N )  ~~  ( `' G `  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289   T. wtru 1290    e. wcel 1438   _Vcvv 2619    u. cun 2995    i^i cin 2996   (/)c0 3284   {csn 3441   class class class wbr 3837    |-> cmpt 3891   Ord word 4180   suc csuc 4183   omcom 4395   `'ccnv 4427    Fn wfn 4997   -1-1-onto->wf1o 5001   ` cfv 5002  (class class class)co 5634  freccfrec 6137    ~~ cen 6435   0cc0 7329   1c1 7330    + caddc 7332    - cmin 7632   NN0cn0 8643   ZZcz 8720   ZZ>=cuz 8988   ...cfz 9393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-recs 6052  df-frec 6138  df-1o 6163  df-er 6272  df-en 6438  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-fz 9394
This theorem is referenced by:  frecfzen2  9799  hashfz1  10156
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