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Theorem ennnfonelemrn 11966
Description: Lemma for ennnfone 11972. 
L is onto  A. (Contributed by Jim Kingdon, 16-Jul-2023.)
Hypotheses
Ref Expression
ennnfonelemh.dceq  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
ennnfonelemh.f  |-  ( ph  ->  F : om -onto-> A
)
ennnfonelemh.ne  |-  ( ph  ->  A. n  e.  om  E. k  e.  om  A. j  e.  suc  n ( F `  k )  =/=  ( F `  j ) )
ennnfonelemh.g  |-  G  =  ( x  e.  ( A  ^pm  om ) ,  y  e.  om  |->  if ( ( F `  y )  e.  ( F " y ) ,  x ,  ( x  u.  { <. dom  x ,  ( F `
 y ) >. } ) ) )
ennnfonelemh.n  |-  N  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
ennnfonelemh.j  |-  J  =  ( x  e.  NN0  |->  if ( x  =  0 ,  (/) ,  ( `' N `  ( x  -  1 ) ) ) )
ennnfonelemh.h  |-  H  =  seq 0 ( G ,  J )
ennnfone.l  |-  L  = 
U_ i  e.  NN0  ( H `  i )
Assertion
Ref Expression
ennnfonelemrn  |-  ( ph  ->  ran  L  =  A )
Distinct variable groups:    A, j, x, y    i, F, j, x, y, k    n, F, k    j, G    i, H, j, x, y, k   
j, J    i, N, j, x, y, k    ph, i,
j, x, y, k   
j, n
Allowed substitution hints:    ph( n)    A( i,
k, n)    G( x, y, i, k, n)    H( n)    J( x, y, i, k, n)    L( x, y, i, j, k, n)    N( n)

Proof of Theorem ennnfonelemrn
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ennnfonelemh.dceq . . . 4  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
2 ennnfonelemh.f . . . 4  |-  ( ph  ->  F : om -onto-> A
)
3 ennnfonelemh.ne . . . 4  |-  ( ph  ->  A. n  e.  om  E. k  e.  om  A. j  e.  suc  n ( F `  k )  =/=  ( F `  j ) )
4 ennnfonelemh.g . . . 4  |-  G  =  ( x  e.  ( A  ^pm  om ) ,  y  e.  om  |->  if ( ( F `  y )  e.  ( F " y ) ,  x ,  ( x  u.  { <. dom  x ,  ( F `
 y ) >. } ) ) )
5 ennnfonelemh.n . . . 4  |-  N  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
6 ennnfonelemh.j . . . 4  |-  J  =  ( x  e.  NN0  |->  if ( x  =  0 ,  (/) ,  ( `' N `  ( x  -  1 ) ) ) )
7 ennnfonelemh.h . . . 4  |-  H  =  seq 0 ( G ,  J )
8 ennnfone.l . . . 4  |-  L  = 
U_ i  e.  NN0  ( H `  i )
91, 2, 3, 4, 5, 6, 7, 8ennnfonelemf1 11965 . . 3  |-  ( ph  ->  L : dom  L -1-1-> A )
10 f1f 5335 . . 3  |-  ( L : dom  L -1-1-> A  ->  L : dom  L --> A )
11 frn 5288 . . 3  |-  ( L : dom  L --> A  ->  ran  L  C_  A )
129, 10, 113syl 17 . 2  |-  ( ph  ->  ran  L  C_  A
)
13 foelrn 5661 . . . . . 6  |-  ( ( F : om -onto-> A  /\  w  e.  A
)  ->  E. j  e.  om  w  =  ( F `  j ) )
142, 13sylan 281 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  E. j  e.  om  w  =  ( F `  j ) )
15 0zd 9089 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  0  e.  ZZ )
16 simprl 521 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  j  e.  om )
17 peano2 4516 . . . . . . . . 9  |-  ( j  e.  om  ->  suc  j  e.  om )
1816, 17syl 14 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  suc  j  e.  om )
1915, 5, 18frec2uzuzd 10205 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  ( N `  suc  j )  e.  (
ZZ>= `  0 ) )
20 nn0uz 9383 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2119, 20eleqtrrdi 2234 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  ( N `  suc  j )  e.  NN0 )
22 fofn 5354 . . . . . . . . . 10  |-  ( F : om -onto-> A  ->  F  Fn  om )
232, 22syl 14 . . . . . . . . 9  |-  ( ph  ->  F  Fn  om )
2423ad2antrr 480 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  F  Fn  om )
25 ordom 4527 . . . . . . . . 9  |-  Ord  om
26 ordsucss 4427 . . . . . . . . 9  |-  ( Ord 
om  ->  ( j  e. 
om  ->  suc  j  C_  om ) )
2725, 16, 26mpsyl 65 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  suc  j  C_  om )
28 vex 2692 . . . . . . . . . 10  |-  j  e. 
_V
2928sucid 4346 . . . . . . . . 9  |-  j  e. 
suc  j
3029a1i 9 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  j  e.  suc  j )
31 fnfvima 5659 . . . . . . . 8  |-  ( ( F  Fn  om  /\  suc  j  C_  om  /\  j  e.  suc  j )  ->  ( F `  j )  e.  ( F " suc  j
) )
3224, 27, 30, 31syl3anc 1217 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  ( F `  j )  e.  ( F " suc  j
) )
33 simprr 522 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  w  =  ( F `  j ) )
3415, 5frec2uzf1od 10209 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  N : om -1-1-onto-> ( ZZ>=
`  0 ) )
35 f1ocnvfv1 5685 . . . . . . . . 9  |-  ( ( N : om -1-1-onto-> ( ZZ>= `  0 )  /\  suc  j  e.  om )  ->  ( `' N `  ( N `  suc  j ) )  =  suc  j )
3634, 18, 35syl2anc 409 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  ( `' N `  ( N `  suc  j ) )  =  suc  j )
3736imaeq2d 4888 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  ( F "
( `' N `  ( N `  suc  j
) ) )  =  ( F " suc  j ) )
3832, 33, 373eltr4d 2224 . . . . . 6  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  w  e.  ( F " ( `' N `  ( N `
 suc  j )
) ) )
39 fveq2 5428 . . . . . . . . 9  |-  ( i  =  ( N `  suc  j )  ->  ( `' N `  i )  =  ( `' N `  ( N `  suc  j ) ) )
4039imaeq2d 4888 . . . . . . . 8  |-  ( i  =  ( N `  suc  j )  ->  ( F " ( `' N `  i ) )  =  ( F " ( `' N `  ( N `
 suc  j )
) ) )
4140eleq2d 2210 . . . . . . 7  |-  ( i  =  ( N `  suc  j )  ->  (
w  e.  ( F
" ( `' N `  i ) )  <->  w  e.  ( F " ( `' N `  ( N `
 suc  j )
) ) ) )
4241rspcev 2792 . . . . . 6  |-  ( ( ( N `  suc  j )  e.  NN0  /\  w  e.  ( F
" ( `' N `  ( N `  suc  j ) ) ) )  ->  E. i  e.  NN0  w  e.  ( F " ( `' N `  i ) ) )
4321, 38, 42syl2anc 409 . . . . 5  |-  ( ( ( ph  /\  w  e.  A )  /\  (
j  e.  om  /\  w  =  ( F `  j ) ) )  ->  E. i  e.  NN0  w  e.  ( F " ( `' N `  i ) ) )
4414, 43rexlimddv 2557 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  E. i  e.  NN0  w  e.  ( F " ( `' N `  i ) ) )
45 eliun 3824 . . . 4  |-  ( w  e.  U_ i  e. 
NN0  ( F "
( `' N `  i ) )  <->  E. i  e.  NN0  w  e.  ( F " ( `' N `  i ) ) )
4644, 45sylibr 133 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  U_ i  e.  NN0  ( F " ( `' N `  i ) ) )
478rneqi 4774 . . . . . . 7  |-  ran  L  =  ran  U_ i  e.  NN0  ( H `  i )
48 rniun 4956 . . . . . . 7  |-  ran  U_ i  e.  NN0  ( H `  i )  =  U_ i  e.  NN0  ran  ( H `  i )
4947, 48eqtri 2161 . . . . . 6  |-  ran  L  =  U_ i  e.  NN0  ran  ( H `  i
)
501adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  NN0 )  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y
)
512adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  NN0 )  ->  F : om -onto-> A )
523adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  NN0 )  ->  A. n  e.  om  E. k  e. 
om  A. j  e.  suc  n ( F `  k )  =/=  ( F `  j )
)
53 simpr 109 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  NN0 )  ->  i  e.  NN0 )
5450, 51, 52, 4, 5, 6, 7, 53ennnfonelemhf1o 11960 . . . . . . . 8  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( H `  i ) : dom  ( H `  i ) -1-1-onto-> ( F " ( `' N `  i ) ) )
55 f1ofo 5381 . . . . . . . 8  |-  ( ( H `  i ) : dom  ( H `
 i ) -1-1-onto-> ( F
" ( `' N `  i ) )  -> 
( H `  i
) : dom  ( H `  i ) -onto->
( F " ( `' N `  i ) ) )
56 forn 5355 . . . . . . . 8  |-  ( ( H `  i ) : dom  ( H `
 i ) -onto-> ( F " ( `' N `  i ) )  ->  ran  ( H `
 i )  =  ( F " ( `' N `  i ) ) )
5754, 55, 563syl 17 . . . . . . 7  |-  ( (
ph  /\  i  e.  NN0 )  ->  ran  ( H `
 i )  =  ( F " ( `' N `  i ) ) )
5857iuneq2dv 3841 . . . . . 6  |-  ( ph  ->  U_ i  e.  NN0  ran  ( H `  i
)  =  U_ i  e.  NN0  ( F "
( `' N `  i ) ) )
5949, 58syl5eq 2185 . . . . 5  |-  ( ph  ->  ran  L  =  U_ i  e.  NN0  ( F
" ( `' N `  i ) ) )
6059eleq2d 2210 . . . 4  |-  ( ph  ->  ( w  e.  ran  L  <-> 
w  e.  U_ i  e.  NN0  ( F "
( `' N `  i ) ) ) )
6160adantr 274 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
w  e.  ran  L  <->  w  e.  U_ i  e. 
NN0  ( F "
( `' N `  i ) ) ) )
6246, 61mpbird 166 . 2  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  ran  L )
6312, 62eqelssd 3120 1  |-  ( ph  ->  ran  L  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 820    = wceq 1332    e. wcel 1481    =/= wne 2309   A.wral 2417   E.wrex 2418    u. cun 3073    C_ wss 3075   (/)c0 3367   ifcif 3478   {csn 3531   <.cop 3534   U_ciun 3820    |-> cmpt 3996   Ord word 4291   suc csuc 4294   omcom 4511   `'ccnv 4545   dom cdm 4546   ran crn 4547   "cima 4549    Fn wfn 5125   -->wf 5126   -1-1->wf1 5127   -onto->wfo 5128   -1-1-onto->wf1o 5129   ` cfv 5130  (class class class)co 5781    e. cmpo 5783  freccfrec 6294    ^pm cpm 6550   0cc0 7643   1c1 7644    + caddc 7646    - cmin 7956   NN0cn0 9000   ZZcz 9077   ZZ>=cuz 9349    seqcseq 10248
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-ltadd 7759
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-pm 6552  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-inn 8744  df-n0 9001  df-z 9078  df-uz 9350  df-seqfrec 10249
This theorem is referenced by:  ennnfonelemen  11968
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