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Theorem fin23lem22 8707
Description: Lemma for fin23 8769 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 8708) 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 8706 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  i
)
3 riotacl 6260 . . 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 753 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  S  C_  om )
6 simpr 461 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  a  e.  S )
75, 6sseldd 3505 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  a  e.  om )
8 nnfi 7710 . . 3  |-  ( a  e.  om  ->  a  e.  Fin )
9 infi 7743 . . 3  |-  ( a  e.  Fin  ->  (
a  i^i  S )  e.  Fin )
10 ficardom 8342 . . 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 8344 . . . . . . 7  |-  ( i  e.  om  ->  ( card `  i )  =  i )
1312eqcomd 2475 . . . . . 6  |-  ( i  e.  om  ->  i  =  ( card `  i
) )
1413eqeq1d 2469 . . . . 5  |-  ( i  e.  om  ->  (
i  =  ( card `  ( a  i^i  S
) )  <->  ( card `  i )  =  (
card `  ( a  i^i  S ) ) ) )
15 eqcom 2476 . . . . 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 727 . . 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 753 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  ->  S  C_  om )
19 simprr 756 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  S )
2018, 19sseldd 3505 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  om )
21 nnon 6690 . . . . . 6  |-  ( a  e.  om  ->  a  e.  On )
22 onenon 8330 . . . . . 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 3718 . . . . 5  |-  ( a  i^i  S )  C_  a
25 ssnum 8420 . . . . 5  |-  ( ( a  e.  dom  card  /\  ( a  i^i  S
)  C_  a )  ->  ( a  i^i  S
)  e.  dom  card )
2623, 24, 25sylancl 662 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
( a  i^i  S
)  e.  dom  card )
27 nnon 6690 . . . . . 6  |-  ( i  e.  om  ->  i  e.  On )
2827ad2antrl 727 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
i  e.  On )
29 onenon 8330 . . . . 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 8368 . . . 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 661 . . 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 716 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  ->  E! j  e.  S  ( j  i^i  S
)  ~~  i )
34 ineq1 3693 . . . . . . 7  |-  ( j  =  a  ->  (
j  i^i  S )  =  ( a  i^i 
S ) )
3534breq1d 4457 . . . . . 6  |-  ( j  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( a  i^i  S )  ~~  i
) )
3635riota2 6268 . . . . 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 661 . . . 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 2476 . . . 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 6511 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 369    = wceq 1379    e. wcel 1767   E!wreu 2816    i^i cin 3475    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   Oncon0 4878   dom cdm 4999   -1-1-onto->wf1o 5587   ` cfv 5588   iota_crio 6244   omcom 6684    ~~ cen 7513   Fincfn 7516   cardccrd 8316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-om 6685  df-recs 7042  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320
This theorem is referenced by:  fin23lem27  8708  fin23lem28  8720  fin23lem30  8722  isf32lem6  8738  isf32lem7  8739  isf32lem8  8740
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