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Theorem fin23lem22 8212
Description: Lemma for fin23 8274 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 8213) 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 8211 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  i
)
3 riotacl 6567 . . 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 732 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  S  C_  om )
6 simpr 449 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  a  e.  S )
75, 6sseldd 3351 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  a  e.  om )
8 nnfi 7302 . . 3  |-  ( a  e.  om  ->  a  e.  Fin )
9 infi 7335 . . 3  |-  ( a  e.  Fin  ->  (
a  i^i  S )  e.  Fin )
10 ficardom 7853 . . 3  |-  ( ( a  i^i  S )  e.  Fin  ->  ( card `  ( a  i^i 
S ) )  e. 
om )
117, 8, 9, 104syl 20 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  ( card `  ( a  i^i  S
) )  e.  om )
12 cardnn 7855 . . . . . . 7  |-  ( i  e.  om  ->  ( card `  i )  =  i )
1312eqcomd 2443 . . . . . 6  |-  ( i  e.  om  ->  i  =  ( card `  i
) )
1413eqeq1d 2446 . . . . 5  |-  ( i  e.  om  ->  (
i  =  ( card `  ( a  i^i  S
) )  <->  ( card `  i )  =  (
card `  ( a  i^i  S ) ) ) )
15 eqcom 2440 . . . . 5  |-  ( (
card `  i )  =  ( card `  (
a  i^i  S )
)  <->  ( card `  (
a  i^i  S )
)  =  ( card `  i ) )
1614, 15syl6bb 254 . . . 4  |-  ( i  e.  om  ->  (
i  =  ( card `  ( a  i^i  S
) )  <->  ( card `  ( a  i^i  S
) )  =  (
card `  i )
) )
1716ad2antrl 710 . . 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 732 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  ->  S  C_  om )
19 simprr 735 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  S )
2018, 19sseldd 3351 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  om )
21 nnon 4854 . . . . . 6  |-  ( a  e.  om  ->  a  e.  On )
22 onenon 7841 . . . . . 6  |-  ( a  e.  On  ->  a  e.  dom  card )
2320, 21, 223syl 19 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  dom  card )
24 inss1 3563 . . . . 5  |-  ( a  i^i  S )  C_  a
25 ssnum 7925 . . . . 5  |-  ( ( a  e.  dom  card  /\  ( a  i^i  S
)  C_  a )  ->  ( a  i^i  S
)  e.  dom  card )
2623, 24, 25sylancl 645 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
( a  i^i  S
)  e.  dom  card )
27 nnon 4854 . . . . . 6  |-  ( i  e.  om  ->  i  e.  On )
2827ad2antrl 710 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
i  e.  On )
29 onenon 7841 . . . . 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 7879 . . . 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 644 . . 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 699 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  ->  E! j  e.  S  ( j  i^i  S
)  ~~  i )
34 ineq1 3537 . . . . . . 7  |-  ( j  =  a  ->  (
j  i^i  S )  =  ( a  i^i 
S ) )
3534breq1d 4225 . . . . . 6  |-  ( j  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( a  i^i  S )  ~~  i
) )
3635riota2 6575 . . . . 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 644 . . . 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 2440 . . . 4  |-  ( (
iota_ j  e.  S
( j  i^i  S
)  ~~  i )  =  a  <->  a  =  (
iota_ j  e.  S
( j  i^i  S
)  ~~  i )
)
3937, 38syl6bb 254 . . 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 272 . 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 6299 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   E!wreu 2709    i^i cin 3321    C_ wss 3322   class class class wbr 4215    e. cmpt 4269   Oncon0 4584   omcom 4848   dom cdm 4881   -1-1-onto->wf1o 5456   ` cfv 5457   iota_crio 6545    ~~ cen 7109   Fincfn 7112   cardccrd 7827
This theorem is referenced by:  fin23lem27  8213  fin23lem28  8225  fin23lem30  8227  isf32lem6  8243  isf32lem7  8244  isf32lem8  8245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-riota 6552  df-recs 6636  df-1o 6727  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831
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