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Theorem fihashdom 10716
Description: Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)
Assertion
Ref Expression
fihashdom  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( `  A
)  <_  ( `  B
)  <->  A  ~<_  B )
)

Proof of Theorem fihashdom
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6727 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32adantr 274 . 2  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  E. n  e.  om  A  ~~  n )
4 isfi 6727 . . . . 5  |-  ( B  e.  Fin  <->  E. m  e.  om  B  ~~  m
)
54biimpi 119 . . . 4  |-  ( B  e.  Fin  ->  E. m  e.  om  B  ~~  m
)
65ad2antlr 481 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  E. m  e.  om  B  ~~  m
)
7 simplrr 526 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  A  ~~  n )
8 domen1 6808 . . . . 5  |-  ( A 
~~  n  ->  ( A  ~<_  m  <->  n  ~<_  m ) )
97, 8syl 14 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( A  ~<_  m  <->  n  ~<_  m ) )
10 simprr 522 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  B  ~~  m )
11 domen2 6809 . . . . 5  |-  ( B 
~~  m  ->  ( A  ~<_  B  <->  A  ~<_  m ) )
1210, 11syl 14 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( A  ~<_  B  <->  A  ~<_  m ) )
13 0zd 9203 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
0  e.  ZZ )
14 eqid 2165 . . . . . 6  |- frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )
15 simplrl 525 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  n  e.  om )
16 simprl 521 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  m  e.  om )
1713, 14, 15, 16frec2uzled 10364 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( n  C_  m  <->  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 ) `  n
)  <_  (frec (
( x  e.  ZZ  |->  ( x  +  1
) ) ,  0 ) `  m ) ) )
18 nndomo 6830 . . . . . 6  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  ~<_  m  <->  n  C_  m
) )
1915, 16, 18syl2anc 409 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( n  ~<_  m  <->  n  C_  m
) )
207ensymd 6749 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  n  ~~  A )
21 hashennn 10693 . . . . . . 7  |-  ( ( n  e.  om  /\  n  ~~  A )  -> 
( `  A )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n ) )
2215, 20, 21syl2anc 409 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( `  A )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n ) )
2310ensymd 6749 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  ->  m  ~~  B )
24 hashennn 10693 . . . . . . 7  |-  ( ( m  e.  om  /\  m  ~~  B )  -> 
( `  B )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  m ) )
2516, 23, 24syl2anc 409 . . . . . 6  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( `  B )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  m ) )
2622, 25breq12d 3995 . . . . 5  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( ( `  A
)  <_  ( `  B
)  <->  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  n )  <_  (frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 ) `  m
) ) )
2717, 19, 263bitr4rd 220 . . . 4  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( ( `  A
)  <_  ( `  B
)  <->  n  ~<_  m )
)
289, 12, 273bitr4rd 220 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  (
n  e.  om  /\  A  ~~  n ) )  /\  ( m  e. 
om  /\  B  ~~  m ) )  -> 
( ( `  A
)  <_  ( `  B
)  <->  A  ~<_  B )
)
296, 28rexlimddv 2588 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  ( ( `  A )  <_  ( `  B )  <->  A  ~<_  B ) )
303, 29rexlimddv 2588 1  |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( `  A
)  <_  ( `  B
)  <->  A  ~<_  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1343    e. wcel 2136   E.wrex 2445    C_ wss 3116   class class class wbr 3982    |-> cmpt 4043   omcom 4567   ` cfv 5188  (class class class)co 5842  freccfrec 6358    ~~ cen 6704    ~<_ cdom 6705   Fincfn 6706   0cc0 7753   1c1 7754    + caddc 7756    <_ cle 7934   ZZcz 9191  ♯chash 10688
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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-recs 6273  df-frec 6359  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-ihash 10689
This theorem is referenced by:  fihashss  10729  phicl2  12146  phibnd  12149
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