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Theorem ennnfonelemf1 12360
Description: Lemma for ennnfone 12367. 
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 12359 . . . 4  |-  ( ph  ->  Fun  L )
109funfnd 5227 . . 3  |-  ( ph  ->  L  Fn  dom  L
)
111, 2, 3, 4, 5, 6, 7ennnfonelemh 12346 . . . . . . . . 9  |-  ( ph  ->  H : NN0 --> ( A 
^pm  om ) )
1211ffnd 5346 . . . . . . . 8  |-  ( ph  ->  H  Fn  NN0 )
13 fniunfv 5738 . . . . . . . 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, 14eqtrid 2215 . . . . . 6  |-  ( ph  ->  L  =  U. ran  H )
1615rneqd 4838 . . . . 5  |-  ( ph  ->  ran  L  =  ran  U.
ran  H )
17 rnuni 5020 . . . . 5  |-  ran  U. ran  H  =  U_ x  e.  ran  H ran  x
1816, 17eqtrdi 2219 . . . 4  |-  ( ph  ->  ran  L  =  U_ x  e.  ran  H ran  x )
1911frnd 5355 . . . . . . . . . 10  |-  ( ph  ->  ran  H  C_  ( A  ^pm  om ) )
2019sselda 3147 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  H )  ->  x  e.  ( A  ^pm  om )
)
21 elpmi 6641 . . . . . . . . 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 5355 . . . . . 6  |-  ( (
ph  /\  x  e.  ran  H )  ->  ran  x  C_  A )
2524ralrimiva 2543 . . . . 5  |-  ( ph  ->  A. x  e.  ran  H ran  x  C_  A
)
26 iunss 3912 . . . . 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 3183 . . 3  |-  ( ph  ->  ran  L  C_  A
)
29 df-f 5200 . . 3  |-  ( L : dom  L --> A  <->  ( L  Fn  dom  L  /\  ran  L 
C_  A ) )
3010, 28, 29sylanbrc 415 . 2  |-  ( ph  ->  L : dom  L --> A )
3119sselda 3147 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ran  H )  ->  s  e.  ( A  ^pm  om )
)
32 pmfun 6642 . . . . . . . 8  |-  ( s  e.  ( A  ^pm  om )  ->  Fun  s )
3331, 32syl 14 . . . . . . 7  |-  ( (
ph  /\  s  e.  ran  H )  ->  Fun  s )
3411ffund 5349 . . . . . . . . . 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 5632 . . . . . . . . 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 5352 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  H  =  NN0 )
4342eleq2d 2240 . . . . . . . . . . . . 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 12355 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  dom  H )  ->  ( H `  q ) : dom  ( H `  q ) -1-1-onto-> ( F " ( `' N `  q ) ) )
46 f1ocnv 5453 . . . . . . . . . . 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 5442 . . . . . . . . . . 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 506 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  ( q  e.  dom  H  /\  s  =  ( H `  q ) ) )  ->  Fun  `' ( H `  q
) )
50 simprr 527 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  ( q  e.  dom  H  /\  s  =  ( H `  q ) ) )  ->  s  =  ( H `  q ) )
5150cnveqd 4785 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  ( q  e.  dom  H  /\  s  =  ( H `  q ) ) )  ->  `' s  =  `' ( H `  q )
)
5251funeqd 5218 . . . . . . . . 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 2592 . . . . . . 7  |-  ( (
ph  /\  s  e.  ran  H )  ->  Fun  `' s )
551ad2antrr 485 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
562ad2antrr 485 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  F : om -onto-> A )
573ad2antrr 485 . . . . . . . . 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 525 . . . . . . . . 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 12358 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  ran  H )  /\  t  e.  ran  H )  ->  ( s  C_  t  \/  t  C_  s ) )
6160ralrimiva 2543 . . . . . . 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 2543 . . . . 5  |-  ( ph  ->  A. s  e.  ran  H ( ( Fun  s  /\  Fun  `' s )  /\  A. t  e. 
ran  H ( s 
C_  t  \/  t  C_  s ) ) )
64 fun11uni 5266 . . . . 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 4785 . . . 4  |-  ( ph  ->  `' L  =  `' U. ran  H )
6867funeqd 5218 . . 3  |-  ( ph  ->  ( Fun  `' L  <->  Fun  `' U. ran  H ) )
6966, 68mpbird 166 . 2  |-  ( ph  ->  Fun  `' L )
70 df-f1 5201 . 2  |-  ( L : dom  L -1-1-> A  <->  ( L : dom  L --> A  /\  Fun  `' L
) )
7130, 69, 70sylanbrc 415 1  |-  ( ph  ->  L : dom  L -1-1-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 703  DECID wdc 829    = wceq 1348    e. wcel 2141    =/= wne 2340   A.wral 2448   E.wrex 2449    u. cun 3119    C_ wss 3121   (/)c0 3414   ifcif 3525   {csn 3581   <.cop 3584   U.cuni 3794   U_ciun 3871    |-> cmpt 4048   suc csuc 4348   omcom 4572   `'ccnv 4608   dom cdm 4609   ran crn 4610   "cima 4612   Fun wfun 5190    Fn wfn 5191   -->wf 5192   -1-1->wf1 5193   -onto->wfo 5194   -1-1-onto->wf1o 5195   ` cfv 5196  (class class class)co 5850    e. cmpo 5852  freccfrec 6366    ^pm cpm 6623   0cc0 7761   1c1 7762    + caddc 7764    - cmin 8077   NN0cn0 9122   ZZcz 9199    seqcseq 10388
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-ltadd 7877
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-pm 6625  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-seqfrec 10389
This theorem is referenced by:  ennnfonelemrn  12361  ennnfonelemen  12363
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