MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem22 Unicode version

Theorem fin23lem22 8163
Description: Lemma for fin23 8225 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 8164) between an infinite subset of  om and  om itself. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Hypothesis
Ref Expression
fin23lem22.b  |-  C  =  ( i  e.  om  |->  ( iota_ j  e.  S
( j  i^i  S
)  ~~  i )
)
Assertion
Ref Expression
fin23lem22  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -1-1-onto-> S )
Distinct variable group:    i, j, S
Allowed substitution hints:    C( i, j)

Proof of Theorem fin23lem22
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fin23lem22.b . 2  |-  C  =  ( i  e.  om  |->  ( iota_ j  e.  S
( j  i^i  S
)  ~~  i )
)
2 fin23lem23 8162 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  i
)
3 riotacl 6523 . . 3  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  i  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  i )  e.  S )
42, 3syl 16 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  ( iota_ j  e.  S ( j  i^i 
S )  ~~  i
)  e.  S )
5 simpll 731 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  S  C_  om )
6 simpr 448 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  a  e.  S )
75, 6sseldd 3309 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  a  e.  om )
8 nnfi 7258 . . 3  |-  ( a  e.  om  ->  a  e.  Fin )
9 infi 7291 . . . 4  |-  ( a  e.  Fin  ->  (
a  i^i  S )  e.  Fin )
10 ficardom 7804 . . . 4  |-  ( ( a  i^i  S )  e.  Fin  ->  ( card `  ( a  i^i 
S ) )  e. 
om )
119, 10syl 16 . . 3  |-  ( a  e.  Fin  ->  ( card `  ( a  i^i 
S ) )  e. 
om )
127, 8, 113syl 19 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  ( card `  ( a  i^i  S
) )  e.  om )
13 cardnn 7806 . . . . . . 7  |-  ( i  e.  om  ->  ( card `  i )  =  i )
1413eqcomd 2409 . . . . . 6  |-  ( i  e.  om  ->  i  =  ( card `  i
) )
1514eqeq1d 2412 . . . . 5  |-  ( i  e.  om  ->  (
i  =  ( card `  ( a  i^i  S
) )  <->  ( card `  i )  =  (
card `  ( a  i^i  S ) ) ) )
16 eqcom 2406 . . . . 5  |-  ( (
card `  i )  =  ( card `  (
a  i^i  S )
)  <->  ( card `  (
a  i^i  S )
)  =  ( card `  i ) )
1715, 16syl6bb 253 . . . 4  |-  ( i  e.  om  ->  (
i  =  ( card `  ( a  i^i  S
) )  <->  ( card `  ( a  i^i  S
) )  =  (
card `  i )
) )
1817ad2antrl 709 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
( i  =  (
card `  ( a  i^i  S ) )  <->  ( card `  ( a  i^i  S
) )  =  (
card `  i )
) )
19 simpll 731 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  ->  S  C_  om )
20 simprr 734 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  S )
2119, 20sseldd 3309 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  om )
22 nnon 4810 . . . . . 6  |-  ( a  e.  om  ->  a  e.  On )
23 onenon 7792 . . . . . 6  |-  ( a  e.  On  ->  a  e.  dom  card )
2421, 22, 233syl 19 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  dom  card )
25 inss1 3521 . . . . 5  |-  ( a  i^i  S )  C_  a
26 ssnum 7876 . . . . 5  |-  ( ( a  e.  dom  card  /\  ( a  i^i  S
)  C_  a )  ->  ( a  i^i  S
)  e.  dom  card )
2724, 25, 26sylancl 644 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
( a  i^i  S
)  e.  dom  card )
28 nnon 4810 . . . . . 6  |-  ( i  e.  om  ->  i  e.  On )
2928ad2antrl 709 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
i  e.  On )
30 onenon 7792 . . . . 5  |-  ( i  e.  On  ->  i  e.  dom  card )
3129, 30syl 16 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
i  e.  dom  card )
32 carden2 7830 . . . 4  |-  ( ( ( a  i^i  S
)  e.  dom  card  /\  i  e.  dom  card )  ->  ( ( card `  ( a  i^i  S
) )  =  (
card `  i )  <->  ( a  i^i  S ) 
~~  i ) )
3327, 31, 32syl2anc 643 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
( ( card `  (
a  i^i  S )
)  =  ( card `  i )  <->  ( a  i^i  S )  ~~  i
) )
342adantrr 698 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  ->  E! j  e.  S  ( j  i^i  S
)  ~~  i )
35 ineq1 3495 . . . . . . 7  |-  ( j  =  a  ->  (
j  i^i  S )  =  ( a  i^i 
S ) )
3635breq1d 4182 . . . . . 6  |-  ( j  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( a  i^i  S )  ~~  i
) )
3736riota2 6531 . . . . 5  |-  ( ( a  e.  S  /\  E! j  e.  S  ( j  i^i  S
)  ~~  i )  ->  ( ( a  i^i 
S )  ~~  i  <->  (
iota_ j  e.  S
( j  i^i  S
)  ~~  i )  =  a ) )
3820, 34, 37syl2anc 643 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
( ( a  i^i 
S )  ~~  i  <->  (
iota_ j  e.  S
( j  i^i  S
)  ~~  i )  =  a ) )
39 eqcom 2406 . . . 4  |-  ( (
iota_ j  e.  S
( j  i^i  S
)  ~~  i )  =  a  <->  a  =  (
iota_ j  e.  S
( j  i^i  S
)  ~~  i )
)
4038, 39syl6bb 253 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
( ( a  i^i 
S )  ~~  i  <->  a  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  i
) ) )
4118, 33, 403bitrd 271 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
( i  =  (
card `  ( a  i^i  S ) )  <->  a  =  ( iota_ j  e.  S
( j  i^i  S
)  ~~  i )
) )
421, 4, 12, 41f1o2d 6255 1  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -1-1-onto-> S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E!wreu 2668    i^i cin 3279    C_ wss 3280   class class class wbr 4172    e. cmpt 4226   Oncon0 4541   omcom 4804   dom cdm 4837   -1-1-onto->wf1o 5412   ` cfv 5413   iota_crio 6501    ~~ cen 7065   Fincfn 7068   cardccrd 7778
This theorem is referenced by:  fin23lem27  8164  fin23lem28  8176  fin23lem30  8178  isf32lem6  8194  isf32lem7  8195  isf32lem8  8196
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6508  df-recs 6592  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782
  Copyright terms: Public domain W3C validator