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Theorem fin23lem22 7969
Description: Lemma for fin23 8031 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 7970) 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 7968 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  i
)
3 riotacl 6335 . . 3  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  i  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  i )  e.  S )
42, 3syl 15 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  i  e.  om )  ->  ( iota_ j  e.  S ( j  i^i 
S )  ~~  i
)  e.  S )
5 simpll 730 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  S  C_  om )
6 simpr 447 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  a  e.  S )
75, 6sseldd 3194 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  a  e.  om )
8 nnfi 7069 . . 3  |-  ( a  e.  om  ->  a  e.  Fin )
9 inss1 3402 . . . . 5  |-  ( a  i^i  S )  C_  a
10 ssfi 7099 . . . . 5  |-  ( ( a  e.  Fin  /\  ( a  i^i  S
)  C_  a )  ->  ( a  i^i  S
)  e.  Fin )
119, 10mpan2 652 . . . 4  |-  ( a  e.  Fin  ->  (
a  i^i  S )  e.  Fin )
12 ficardom 7610 . . . 4  |-  ( ( a  i^i  S )  e.  Fin  ->  ( card `  ( a  i^i 
S ) )  e. 
om )
1311, 12syl 15 . . 3  |-  ( a  e.  Fin  ->  ( card `  ( a  i^i 
S ) )  e. 
om )
147, 8, 133syl 18 . 2  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  S
)  ->  ( card `  ( a  i^i  S
) )  e.  om )
15 cardnn 7612 . . . . . . 7  |-  ( i  e.  om  ->  ( card `  i )  =  i )
1615eqcomd 2301 . . . . . 6  |-  ( i  e.  om  ->  i  =  ( card `  i
) )
1716eqeq1d 2304 . . . . 5  |-  ( i  e.  om  ->  (
i  =  ( card `  ( a  i^i  S
) )  <->  ( card `  i )  =  (
card `  ( a  i^i  S ) ) ) )
18 eqcom 2298 . . . . 5  |-  ( (
card `  i )  =  ( card `  (
a  i^i  S )
)  <->  ( card `  (
a  i^i  S )
)  =  ( card `  i ) )
1917, 18syl6bb 252 . . . 4  |-  ( i  e.  om  ->  (
i  =  ( card `  ( a  i^i  S
) )  <->  ( card `  ( a  i^i  S
) )  =  (
card `  i )
) )
2019ad2antrl 708 . . 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 )
) )
21 simpll 730 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  ->  S  C_  om )
22 simprr 733 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  S )
2321, 22sseldd 3194 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  om )
24 nnon 4678 . . . . . 6  |-  ( a  e.  om  ->  a  e.  On )
25 onenon 7598 . . . . . 6  |-  ( a  e.  On  ->  a  e.  dom  card )
2623, 24, 253syl 18 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
a  e.  dom  card )
27 ssnum 7682 . . . . 5  |-  ( ( a  e.  dom  card  /\  ( a  i^i  S
)  C_  a )  ->  ( a  i^i  S
)  e.  dom  card )
2826, 9, 27sylancl 643 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
( a  i^i  S
)  e.  dom  card )
29 nnon 4678 . . . . . 6  |-  ( i  e.  om  ->  i  e.  On )
3029ad2antrl 708 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
i  e.  On )
31 onenon 7598 . . . . 5  |-  ( i  e.  On  ->  i  e.  dom  card )
3230, 31syl 15 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  -> 
i  e.  dom  card )
33 carden2 7636 . . . 4  |-  ( ( ( a  i^i  S
)  e.  dom  card  /\  i  e.  dom  card )  ->  ( ( card `  ( a  i^i  S
) )  =  (
card `  i )  <->  ( a  i^i  S ) 
~~  i ) )
3428, 32, 33syl2anc 642 . . 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
) )
352adantrr 697 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( i  e. 
om  /\  a  e.  S ) )  ->  E! j  e.  S  ( j  i^i  S
)  ~~  i )
36 ineq1 3376 . . . . . . 7  |-  ( j  =  a  ->  (
j  i^i  S )  =  ( a  i^i 
S ) )
3736breq1d 4049 . . . . . 6  |-  ( j  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( a  i^i  S )  ~~  i
) )
3837riota2 6343 . . . . 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 ) )
3922, 35, 38syl2anc 642 . . . 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 ) )
40 eqcom 2298 . . . 4  |-  ( (
iota_ j  e.  S
( j  i^i  S
)  ~~  i )  =  a  <->  a  =  (
iota_ j  e.  S
( j  i^i  S
)  ~~  i )
)
4139, 40syl6bb 252 . . 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
) ) )
4220, 34, 413bitrd 270 . 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 )
) )
431, 4, 14, 42f1o2d 6085 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   E!wreu 2558    i^i cin 3164    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   Oncon0 4408   omcom 4672   dom cdm 4705   -1-1-onto->wf1o 5270   ` cfv 5271   iota_crio 6313    ~~ cen 6876   Fincfn 6879   cardccrd 7584
This theorem is referenced by:  fin23lem27  7970  fin23lem28  7982  fin23lem30  7984  isf32lem6  8000  isf32lem7  8001  isf32lem8  8002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588
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