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Theorem ennnfonelemf1 11965
Description: Lemma for ennnfone 11972. 
L is one-to-one. (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
ennnfonelemf1  |-  ( ph  ->  L : dom  L -1-1-> A )
Distinct variable groups:    A, j, x, y    x, F, y, j, k    n, F   
j, G    i, H    j, H, x, y, k   
j, J    x, N, y, k, j    ph, j, x, y, k    k, n, j
Allowed substitution hints:    ph( i, n)    A( i, k, n)    F( i)    G( x, y, i, k, n)    H( n)    J( x, y, i, k, n)    L( x, y, i, j, k, n)    N( i, n)

Proof of Theorem ennnfonelemf1
Dummy variables  q  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ennnfonelemh.dceq . . . . 5  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
2 ennnfonelemh.f . . . . 5  |-  ( ph  ->  F : om -onto-> A
)
3 ennnfonelemh.ne . . . . 5  |-  ( ph  ->  A. n  e.  om  E. k  e.  om  A. j  e.  suc  n ( F `  k )  =/=  ( F `  j ) )
4 ennnfonelemh.g . . . . 5  |-  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 . . . . 5  |-  N  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
6 ennnfonelemh.j . . . . 5  |-  J  =  ( x  e.  NN0  |->  if ( x  =  0 ,  (/) ,  ( `' N `  ( x  -  1 ) ) ) )
7 ennnfonelemh.h . . . . 5  |-  H  =  seq 0 ( G ,  J )
8 ennnfone.l . . . . 5  |-  L  = 
U_ i  e.  NN0  ( H `  i )
91, 2, 3, 4, 5, 6, 7, 8ennnfonelemfun 11964 . . . 4  |-  ( ph  ->  Fun  L )
109funfnd 5161 . . 3  |-  ( ph  ->  L  Fn  dom  L
)
111, 2, 3, 4, 5, 6, 7ennnfonelemh 11951 . . . . . . . . 9  |-  ( ph  ->  H : NN0 --> ( A 
^pm  om ) )
1211ffnd 5280 . . . . . . . 8  |-  ( ph  ->  H  Fn  NN0 )
13 fniunfv 5670 . . . . . . . 8  |-  ( H  Fn  NN0  ->  U_ i  e.  NN0  ( H `  i )  =  U. ran  H )
1412, 13syl 14 . . . . . . 7  |-  ( ph  ->  U_ i  e.  NN0  ( H `  i )  =  U. ran  H
)
158, 14syl5eq 2185 . . . . . 6  |-  ( ph  ->  L  =  U. ran  H )
1615rneqd 4775 . . . . 5  |-  ( ph  ->  ran  L  =  ran  U.
ran  H )
17 rnuni 4957 . . . . 5  |-  ran  U. ran  H  =  U_ x  e.  ran  H ran  x
1816, 17eqtrdi 2189 . . . 4  |-  ( ph  ->  ran  L  =  U_ x  e.  ran  H ran  x )
1911frnd 5289 . . . . . . . . . 10  |-  ( ph  ->  ran  H  C_  ( A  ^pm  om ) )
2019sselda 3101 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  H )  ->  x  e.  ( A  ^pm  om )
)
21 elpmi 6568 . . . . . . . . 9  |-  ( x  e.  ( A  ^pm  om )  ->  ( x : dom  x --> A  /\  dom  x  C_  om )
)
2220, 21syl 14 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ran  H )  ->  (
x : dom  x --> A  /\  dom  x  C_  om ) )
2322simpld 111 . . . . . . 7  |-  ( (
ph  /\  x  e.  ran  H )  ->  x : dom  x --> A )
2423frnd 5289 . . . . . 6  |-  ( (
ph  /\  x  e.  ran  H )  ->  ran  x  C_  A )
2524ralrimiva 2508 . . . . 5  |-  ( ph  ->  A. x  e.  ran  H ran  x  C_  A
)
26 iunss 3861 . . . . 5  |-  ( U_ x  e.  ran  H ran  x  C_  A  <->  A. x  e.  ran  H ran  x  C_  A )
2725, 26sylibr 133 . . . 4  |-  ( ph  ->  U_ x  e.  ran  H ran  x  C_  A
)
2818, 27eqsstrd 3137 . . 3  |-  ( ph  ->  ran  L  C_  A
)
29 df-f 5134 . . 3  |-  ( L : dom  L --> A  <->  ( L  Fn  dom  L  /\  ran  L 
C_  A ) )
3010, 28, 29sylanbrc 414 . 2  |-  ( ph  ->  L : dom  L --> A )
3119sselda 3101 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ran  H )  ->  s  e.  ( A  ^pm  om )
)
32 pmfun 6569 . . . . . . . 8  |-  ( s  e.  ( A  ^pm  om )  ->  Fun  s )
3331, 32syl 14 . . . . . . 7  |-  ( (
ph  /\  s  e.  ran  H )  ->  Fun  s )
3411ffund 5283 . . . . . . . . . 10  |-  ( ph  ->  Fun  H )
3534adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ran  H )  ->  Fun  H )
36 simpr 109 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ran  H )  ->  s  e.  ran  H )
37 elrnrexdm 5566 . . . . . . . . 9  |-  ( Fun 
H  ->  ( s  e.  ran  H  ->  E. q  e.  dom  H  s  =  ( H `  q
) ) )
3835, 36, 37sylc 62 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ran  H )  ->  E. q  e.  dom  H  s  =  ( H `  q
) )
391adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  dom  H )  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y
)
402adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  dom  H )  ->  F : om -onto-> A )
413adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  dom  H )  ->  A. n  e.  om  E. k  e. 
om  A. j  e.  suc  n ( F `  k )  =/=  ( F `  j )
)
4211fdmd 5286 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  H  =  NN0 )
4342eleq2d 2210 . . . . . . . . . . . . 13  |-  ( ph  ->  ( q  e.  dom  H  <-> 
q  e.  NN0 )
)
4443biimpa 294 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  dom  H )  ->  q  e.  NN0 )
4539, 40, 41, 4, 5, 6, 7, 44ennnfonelemhf1o 11960 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  dom  H )  ->  ( H `  q ) : dom  ( H `  q ) -1-1-onto-> ( F " ( `' N `  q ) ) )
46 f1ocnv 5387 . . . . . . . . . . 11  |-  ( ( H `  q ) : dom  ( H `
 q ) -1-1-onto-> ( F
" ( `' N `  q ) )  ->  `' ( H `  q ) : ( F " ( `' N `  q ) ) -1-1-onto-> dom  ( H `  q ) )
47 f1ofun 5376 . . . . . . . . . . 11  |-  ( `' ( H `  q
) : ( F
" ( `' N `  q ) ) -1-1-onto-> dom  ( H `  q )  ->  Fun  `' ( H `
 q ) )
4845, 46, 473syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  dom  H )  ->  Fun  `' ( H `  q
) )
4948ad2ant2r 501 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  ( q  e.  dom  H  /\  s  =  ( H `  q ) ) )  ->  Fun  `' ( H `  q
) )
50 simprr 522 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  ( q  e.  dom  H  /\  s  =  ( H `  q ) ) )  ->  s  =  ( H `  q ) )
5150cnveqd 4722 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  ( q  e.  dom  H  /\  s  =  ( H `  q ) ) )  ->  `' s  =  `' ( H `  q )
)
5251funeqd 5152 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  ( q  e.  dom  H  /\  s  =  ( H `  q ) ) )  ->  ( Fun  `' s  <->  Fun  `' ( H `
 q ) ) )
5349, 52mpbird 166 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  ( q  e.  dom  H  /\  s  =  ( H `  q ) ) )  ->  Fun  `' s )
5438, 53rexlimddv 2557 . . . . . . 7  |-  ( (
ph  /\  s  e.  ran  H )  ->  Fun  `' s )
551ad2antrr 480 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
562ad2antrr 480 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  F : om -onto-> A )
573ad2antrr 480 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  A. n  e.  om  E. k  e.  om  A. j  e.  suc  n ( F `  k )  =/=  ( F `  j ) )
58 simplr 520 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  s  e.  ran  H )
59 simpr 109 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  t  e.  ran  H )
6055, 56, 57, 4, 5, 6, 7, 58, 59ennnfonelemrnh 11963 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  ( s  C_  t  \/  t  C_  s ) )
6160ralrimiva 2508 . . . . . . 7  |-  ( (
ph  /\  s  e.  ran  H )  ->  A. t  e.  ran  H ( s 
C_  t  \/  t  C_  s ) )
6233, 54, 61jca31 307 . . . . . 6  |-  ( (
ph  /\  s  e.  ran  H )  ->  (
( Fun  s  /\  Fun  `' s )  /\  A. t  e.  ran  H
( s  C_  t  \/  t  C_  s ) ) )
6362ralrimiva 2508 . . . . 5  |-  ( ph  ->  A. s  e.  ran  H ( ( Fun  s  /\  Fun  `' s )  /\  A. t  e. 
ran  H ( s 
C_  t  \/  t  C_  s ) ) )
64 fun11uni 5200 . . . . 5  |-  ( A. s  e.  ran  H ( ( Fun  s  /\  Fun  `' s )  /\  A. t  e.  ran  H
( s  C_  t  \/  t  C_  s ) )  ->  ( Fun  U.
ran  H  /\  Fun  `' U. ran  H ) )
6563, 64syl 14 . . . 4  |-  ( ph  ->  ( Fun  U. ran  H  /\  Fun  `' U. ran  H ) )
6665simprd 113 . . 3  |-  ( ph  ->  Fun  `' U. ran  H )
6715cnveqd 4722 . . . 4  |-  ( ph  ->  `' L  =  `' U. ran  H )
6867funeqd 5152 . . 3  |-  ( ph  ->  ( Fun  `' L  <->  Fun  `' U. ran  H ) )
6966, 68mpbird 166 . 2  |-  ( ph  ->  Fun  `' L )
70 df-f1 5135 . 2  |-  ( L : dom  L -1-1-> A  <->  ( L : dom  L --> A  /\  Fun  `' L
) )
7130, 69, 70sylanbrc 414 1  |-  ( ph  ->  L : dom  L -1-1-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 698  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.cuni 3743   U_ciun 3820    |-> cmpt 3996   suc csuc 4294   omcom 4511   `'ccnv 4545   dom cdm 4546   ran crn 4547   "cima 4549   Fun wfun 5124    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    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:  ennnfonelemrn  11966  ennnfonelemen  11968
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