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Theorem fin23lem22 8698
Description: Lemma for fin23 8760 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 8699) 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 8697 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  i
)
3 riotacl 6246 . . 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 751 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  S  C_  om )
6 simpr 459 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  a  e.  S )
75, 6sseldd 3490 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  a  e.  om )
8 nnfi 7703 . . 3  |-  ( a  e.  om  ->  a  e.  Fin )
9 infi 7736 . . 3  |-  ( a  e.  Fin  ->  (
a  i^i  S )  e.  Fin )
10 ficardom 8333 . . 3  |-  ( ( a  i^i  S )  e.  Fin  ->  ( card `  ( a  i^i 
S ) )  e. 
om )
117, 8, 9, 104syl 21 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  ( card `  ( a  i^i  S
) )  e.  om )
12 cardnn 8335 . . . . . . 7  |-  ( i  e.  om  ->  ( card `  i )  =  i )
1312eqcomd 2462 . . . . . 6  |-  ( i  e.  om  ->  i  =  ( card `  i
) )
1413eqeq1d 2456 . . . . 5  |-  ( i  e.  om  ->  (
i  =  ( card `  ( a  i^i  S
) )  <->  ( card `  i )  =  (
card `  ( a  i^i  S ) ) ) )
15 eqcom 2463 . . . . 5  |-  ( (
card `  i )  =  ( card `  (
a  i^i  S )
)  <->  ( card `  (
a  i^i  S )
)  =  ( card `  i ) )
1614, 15syl6bb 261 . . . 4  |-  ( i  e.  om  ->  (
i  =  ( card `  ( a  i^i  S
) )  <->  ( card `  ( a  i^i  S
) )  =  (
card `  i )
) )
1716ad2antrl 725 . . 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 )
) )
18 simpll 751 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  ->  S  C_  om )
19 simprr 755 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  S )
2018, 19sseldd 3490 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  om )
21 nnon 6679 . . . . . 6  |-  ( a  e.  om  ->  a  e.  On )
22 onenon 8321 . . . . . 6  |-  ( a  e.  On  ->  a  e.  dom  card )
2320, 21, 223syl 20 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  dom  card )
24 inss1 3704 . . . . 5  |-  ( a  i^i  S )  C_  a
25 ssnum 8411 . . . . 5  |-  ( ( a  e.  dom  card  /\  ( a  i^i  S
)  C_  a )  ->  ( a  i^i  S
)  e.  dom  card )
2623, 24, 25sylancl 660 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
( a  i^i  S
)  e.  dom  card )
27 nnon 6679 . . . . . 6  |-  ( i  e.  om  ->  i  e.  On )
2827ad2antrl 725 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
i  e.  On )
29 onenon 8321 . . . . 5  |-  ( i  e.  On  ->  i  e.  dom  card )
3028, 29syl 16 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
i  e.  dom  card )
31 carden2 8359 . . . 4  |-  ( ( ( a  i^i  S
)  e.  dom  card  /\  i  e.  dom  card )  ->  ( ( card `  ( a  i^i  S
) )  =  (
card `  i )  <->  ( a  i^i  S ) 
~~  i ) )
3226, 30, 31syl2anc 659 . . 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
) )
332adantrr 714 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  ->  E! j  e.  S  ( j  i^i  S
)  ~~  i )
34 ineq1 3679 . . . . . . 7  |-  ( j  =  a  ->  (
j  i^i  S )  =  ( a  i^i 
S ) )
3534breq1d 4449 . . . . . 6  |-  ( j  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( a  i^i  S )  ~~  i
) )
3635riota2 6254 . . . . 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 ) )
3719, 33, 36syl2anc 659 . . . 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 ) )
38 eqcom 2463 . . . 4  |-  ( (
iota_ j  e.  S  ( j  i^i  S
)  ~~  i )  =  a  <->  a  =  (
iota_ j  e.  S  ( j  i^i  S
)  ~~  i )
)
3937, 38syl6bb 261 . . 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
) ) )
4017, 32, 393bitrd 279 . 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 )
) )
411, 4, 11, 40f1o2d 6500 1  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -1-1-onto-> S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E!wreu 2806    i^i cin 3460    C_ wss 3461   class class class wbr 4439    |-> cmpt 4497   Oncon0 4867   dom cdm 4988   -1-1-onto->wf1o 5569   ` cfv 5570   iota_crio 6231   omcom 6673    ~~ cen 7506   Fincfn 7509   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-om 6674  df-recs 7034  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311
This theorem is referenced by:  fin23lem27  8699  fin23lem28  8711  fin23lem30  8713  isf32lem6  8729  isf32lem7  8730  isf32lem8  8731
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