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Theorem ennnfonelemf1 11776
Description: Lemma for ennnfone 11783. 
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 11775 . . . 4  |-  ( ph  ->  Fun  L )
109funfnd 5112 . . 3  |-  ( ph  ->  L  Fn  dom  L
)
111, 2, 3, 4, 5, 6, 7ennnfonelemh 11762 . . . . . . . . 9  |-  ( ph  ->  H : NN0 --> ( A 
^pm  om ) )
1211ffnd 5231 . . . . . . . 8  |-  ( ph  ->  H  Fn  NN0 )
13 fniunfv 5617 . . . . . . . 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 2159 . . . . . 6  |-  ( ph  ->  L  =  U. ran  H )
1615rneqd 4728 . . . . 5  |-  ( ph  ->  ran  L  =  ran  U.
ran  H )
17 rnuni 4908 . . . . 5  |-  ran  U. ran  H  =  U_ x  e.  ran  H ran  x
1816, 17syl6eq 2163 . . . 4  |-  ( ph  ->  ran  L  =  U_ x  e.  ran  H ran  x )
1911frnd 5240 . . . . . . . . . 10  |-  ( ph  ->  ran  H  C_  ( A  ^pm  om ) )
2019sselda 3063 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  H )  ->  x  e.  ( A  ^pm  om )
)
21 elpmi 6515 . . . . . . . . 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 5240 . . . . . 6  |-  ( (
ph  /\  x  e.  ran  H )  ->  ran  x  C_  A )
2524ralrimiva 2479 . . . . 5  |-  ( ph  ->  A. x  e.  ran  H ran  x  C_  A
)
26 iunss 3820 . . . . 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 3099 . . 3  |-  ( ph  ->  ran  L  C_  A
)
29 df-f 5085 . . 3  |-  ( L : dom  L --> A  <->  ( L  Fn  dom  L  /\  ran  L 
C_  A ) )
3010, 28, 29sylanbrc 411 . 2  |-  ( ph  ->  L : dom  L --> A )
3119sselda 3063 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ran  H )  ->  s  e.  ( A  ^pm  om )
)
32 pmfun 6516 . . . . . . . 8  |-  ( s  e.  ( A  ^pm  om )  ->  Fun  s )
3331, 32syl 14 . . . . . . 7  |-  ( (
ph  /\  s  e.  ran  H )  ->  Fun  s )
3411ffund 5234 . . . . . . . . . 10  |-  ( ph  ->  Fun  H )
3534adantr 272 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ran  H )  ->  Fun  H )
36 simpr 109 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ran  H )  ->  s  e.  ran  H )
37 elrnrexdm 5513 . . . . . . . . 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 272 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  dom  H )  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y
)
402adantr 272 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  dom  H )  ->  F : om -onto-> A )
413adantr 272 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  dom  H )  ->  A. n  e.  om  E. k  e. 
om  A. j  e.  suc  n ( F `  k )  =/=  ( F `  j )
)
4211fdmd 5237 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  H  =  NN0 )
4342eleq2d 2184 . . . . . . . . . . . . 13  |-  ( ph  ->  ( q  e.  dom  H  <-> 
q  e.  NN0 )
)
4443biimpa 292 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  dom  H )  ->  q  e.  NN0 )
4539, 40, 41, 4, 5, 6, 7, 44ennnfonelemhf1o 11771 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  dom  H )  ->  ( H `  q ) : dom  ( H `  q ) -1-1-onto-> ( F " ( `' N `  q ) ) )
46 f1ocnv 5336 . . . . . . . . . . 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 5325 . . . . . . . . . . 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 498 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  ( q  e.  dom  H  /\  s  =  ( H `  q ) ) )  ->  Fun  `' ( H `  q
) )
50 simprr 504 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  ( q  e.  dom  H  /\  s  =  ( H `  q ) ) )  ->  s  =  ( H `  q ) )
5150cnveqd 4675 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  ( q  e.  dom  H  /\  s  =  ( H `  q ) ) )  ->  `' s  =  `' ( H `  q )
)
5251funeqd 5103 . . . . . . . . 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 2528 . . . . . . 7  |-  ( (
ph  /\  s  e.  ran  H )  ->  Fun  `' s )
551ad2antrr 477 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
562ad2antrr 477 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  F : om -onto-> A )
573ad2antrr 477 . . . . . . . . 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 502 . . . . . . . . 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 11774 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  ( s  C_  t  \/  t  C_  s ) )
6160ralrimiva 2479 . . . . . . 7  |-  ( (
ph  /\  s  e.  ran  H )  ->  A. t  e.  ran  H ( s 
C_  t  \/  t  C_  s ) )
6233, 54, 61jca31 305 . . . . . 6  |-  ( (
ph  /\  s  e.  ran  H )  ->  (
( Fun  s  /\  Fun  `' s )  /\  A. t  e.  ran  H
( s  C_  t  \/  t  C_  s ) ) )
6362ralrimiva 2479 . . . . 5  |-  ( ph  ->  A. s  e.  ran  H ( ( Fun  s  /\  Fun  `' s )  /\  A. t  e. 
ran  H ( s 
C_  t  \/  t  C_  s ) ) )
64 fun11uni 5151 . . . . 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 4675 . . . 4  |-  ( ph  ->  `' L  =  `' U. ran  H )
6867funeqd 5103 . . 3  |-  ( ph  ->  ( Fun  `' L  <->  Fun  `' U. ran  H ) )
6966, 68mpbird 166 . 2  |-  ( ph  ->  Fun  `' L )
70 df-f1 5086 . 2  |-  ( L : dom  L -1-1-> A  <->  ( L : dom  L --> A  /\  Fun  `' L
) )
7130, 69, 70sylanbrc 411 1  |-  ( ph  ->  L : dom  L -1-1-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680  DECID wdc 802    = wceq 1314    e. wcel 1463    =/= wne 2282   A.wral 2390   E.wrex 2391    u. cun 3035    C_ wss 3037   (/)c0 3329   ifcif 3440   {csn 3493   <.cop 3496   U.cuni 3702   U_ciun 3779    |-> cmpt 3949   suc csuc 4247   omcom 4464   `'ccnv 4498   dom cdm 4499   ran crn 4500   "cima 4502   Fun wfun 5075    Fn wfn 5076   -->wf 5077   -1-1->wf1 5078   -onto->wfo 5079   -1-1-onto->wf1o 5080   ` cfv 5081  (class class class)co 5728    e. cmpo 5730  freccfrec 6241    ^pm cpm 6497   0cc0 7547   1c1 7548    + caddc 7550    - cmin 7856   NN0cn0 8881   ZZcz 8958    seqcseq 10111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-pm 6499  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959  df-uz 9229  df-seqfrec 10112
This theorem is referenced by:  ennnfonelemrn  11777  ennnfonelemen  11779
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