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Theorem fin23lem27 8213
Description: The mapping constructed in fin23lem22 8212 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypothesis
Ref Expression
fin23lem22.b  |-  C  =  ( i  e.  om  |->  ( iota_ j  e.  S
( j  i^i  S
)  ~~  i )
)
Assertion
Ref Expression
fin23lem27  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C  Isom  _E  ,  _E  ( om ,  S ) )
Distinct variable group:    i, j, S
Allowed substitution hints:    C( i, j)

Proof of Theorem fin23lem27
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordom 4857 . . . 4  |-  Ord  om
2 ordwe 4597 . . . 4  |-  ( Ord 
om  ->  _E  We  om )
3 weso 4576 . . . 4  |-  (  _E  We  om  ->  _E  Or  om )
41, 2, 3mp2b 10 . . 3  |-  _E  Or  om
54a1i 11 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  _E  Or  om )
6 sopo 4523 . . . . 5  |-  (  _E  Or  om  ->  _E  Po  om )
74, 6ax-mp 5 . . . 4  |-  _E  Po  om
8 poss 4508 . . . 4  |-  ( S 
C_  om  ->  (  _E  Po  om  ->  _E  Po  S ) )
97, 8mpi 17 . . 3  |-  ( S 
C_  om  ->  _E  Po  S )
109adantr 453 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  _E  Po  S )
11 fin23lem22.b . . . 4  |-  C  =  ( i  e.  om  |->  ( iota_ j  e.  S
( j  i^i  S
)  ~~  i )
)
1211fin23lem22 8212 . . 3  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -1-1-onto-> S )
13 f1ofo 5684 . . 3  |-  ( C : om -1-1-onto-> S  ->  C : om -onto-> S )
1412, 13syl 16 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -onto-> S
)
15 nnsdomel 7882 . . . . . . . 8  |-  ( ( a  e.  om  /\  b  e.  om )  ->  ( a  e.  b  <-> 
a  ~<  b ) )
1615adantl 454 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  <->  a  ~<  b ) )
1716biimpd 200 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
a  ~<  b ) )
18 fin23lem23 8211 . . . . . . . . . . . . 13  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  a
)
1918adantrr 699 . . . . . . . . . . . 12  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! j  e.  S  (
j  i^i  S )  ~~  a )
20 ineq1 3537 . . . . . . . . . . . . . 14  |-  ( j  =  i  ->  (
j  i^i  S )  =  ( i  i^i 
S ) )
2120breq1d 4225 . . . . . . . . . . . . 13  |-  ( j  =  i  ->  (
( j  i^i  S
)  ~~  a  <->  ( i  i^i  S )  ~~  a
) )
2221cbvreuv 2936 . . . . . . . . . . . 12  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  a  <->  E! i  e.  S  ( i  i^i  S )  ~~  a
)
2319, 22sylib 190 . . . . . . . . . . 11  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! i  e.  S  (
i  i^i  S )  ~~  a )
24 nfv 1630 . . . . . . . . . . . 12  |-  F/ i ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~~  a
2521cbvriotav 6564 . . . . . . . . . . . 12  |-  ( iota_ j  e.  S ( j  i^i  S )  ~~  a )  =  (
iota_ i  e.  S
( i  i^i  S
)  ~~  a )
26 ineq1 3537 . . . . . . . . . . . . 13  |-  ( i  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  ->  ( i  i^i  S )  =  ( ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S ) )
2726breq1d 4225 . . . . . . . . . . . 12  |-  ( i  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  ->  ( (
i  i^i  S )  ~~  a  <->  ( ( iota_ j  e.  S ( j  i^i  S )  ~~  a )  i^i  S
)  ~~  a )
)
2824, 25, 27riotaprop 6576 . . . . . . . . . . 11  |-  ( E! i  e.  S  ( i  i^i  S ) 
~~  a  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  S  /\  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S )  ~~  a
) )
2923, 28syl 16 . . . . . . . . . 10  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  S  /\  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S )  ~~  a
) )
3029simprd 451 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S )  ~~  a
)
3130adantrr 699 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~~  a )
32 simprr 735 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
a  ~<  b )
33 fin23lem23 8211 . . . . . . . . . . . . . . 15  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  b  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  b
)
3433adantrl 698 . . . . . . . . . . . . . 14  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! j  e.  S  (
j  i^i  S )  ~~  b )
3520breq1d 4225 . . . . . . . . . . . . . . 15  |-  ( j  =  i  ->  (
( j  i^i  S
)  ~~  b  <->  ( i  i^i  S )  ~~  b
) )
3635cbvreuv 2936 . . . . . . . . . . . . . 14  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  b  <->  E! i  e.  S  ( i  i^i  S )  ~~  b
)
3734, 36sylib 190 . . . . . . . . . . . . 13  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! i  e.  S  (
i  i^i  S )  ~~  b )
38 nfv 1630 . . . . . . . . . . . . . 14  |-  F/ i ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )  ~~  b
3935cbvriotav 6564 . . . . . . . . . . . . . 14  |-  ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  =  (
iota_ i  e.  S
( i  i^i  S
)  ~~  b )
40 ineq1 3537 . . . . . . . . . . . . . . 15  |-  ( i  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  ->  ( i  i^i  S )  =  ( ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S ) )
4140breq1d 4225 . . . . . . . . . . . . . 14  |-  ( i  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  ->  ( (
i  i^i  S )  ~~  b  <->  ( ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  i^i  S
)  ~~  b )
)
4238, 39, 41riotaprop 6576 . . . . . . . . . . . . 13  |-  ( E! i  e.  S  ( i  i^i  S ) 
~~  b  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  e.  S  /\  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S )  ~~  b
) )
4337, 42syl 16 . . . . . . . . . . . 12  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  e.  S  /\  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S )  ~~  b
) )
4443simprd 451 . . . . . . . . . . 11  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S )  ~~  b
)
4544ensymd 7161 . . . . . . . . . 10  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  b  ~~  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )
)
4645adantrr 699 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
b  ~~  ( ( iota_ j  e.  S ( j  i^i  S ) 
~~  b )  i^i 
S ) )
47 sdomentr 7244 . . . . . . . . 9  |-  ( ( a  ~<  b  /\  b  ~~  ( ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  i^i  S
) )  ->  a  ~<  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )
)
4832, 46, 47syl2anc 644 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
a  ~<  ( ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  i^i  S
) )
49 ensdomtr 7246 . . . . . . . 8  |-  ( ( ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~~  a  /\  a  ~<  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )
)  ->  ( ( iota_ j  e.  S ( j  i^i  S ) 
~~  a )  i^i 
S )  ~<  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S ) )
5031, 48, 49syl2anc 644 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~<  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )
)
5150expr 600 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  ~<  b  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S )  ~<  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S ) ) )
52 simpll 732 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  S  C_ 
om )
53 omsson 4852 . . . . . . . . 9  |-  om  C_  On
5452, 53syl6ss 3362 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  S  C_  On )
5529simpld 447 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  a )  e.  S )
5654, 55sseldd 3351 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  a )  e.  On )
5743simpld 447 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  b )  e.  S )
5854, 57sseldd 3351 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  b )  e.  On )
59 onsdominel 7259 . . . . . . . 8  |-  ( ( ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  On  /\  ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  e.  On  /\  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~<  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )
)  ->  ( iota_ j  e.  S ( j  i^i  S )  ~~  a )  e.  (
iota_ j  e.  S
( j  i^i  S
)  ~~  b )
)
60593expia 1156 . . . . . . 7  |-  ( ( ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  On  /\  ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  e.  On )  ->  ( ( (
iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S )  ~<  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S )  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  a )  e.  ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )
) )
6156, 58, 60syl2anc 644 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~<  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )  ->  ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )
) )
6217, 51, 613syld 54 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )
) )
63 simprl 734 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  a  e.  om )
64 breq2 4219 . . . . . . . . 9  |-  ( i  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( j  i^i  S )  ~~  a
) )
6564riotabidv 6554 . . . . . . . 8  |-  ( i  =  a  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  i )  =  ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )
)
6665, 11fvmptg 5807 . . . . . . 7  |-  ( ( a  e.  om  /\  ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  S )  ->  ( C `  a )  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
) )
6763, 55, 66syl2anc 644 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( C `  a )  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
) )
68 simprr 735 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  b  e.  om )
69 breq2 4219 . . . . . . . . 9  |-  ( i  =  b  ->  (
( j  i^i  S
)  ~~  i  <->  ( j  i^i  S )  ~~  b
) )
7069riotabidv 6554 . . . . . . . 8  |-  ( i  =  b  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  i )  =  ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )
)
7170, 11fvmptg 5807 . . . . . . 7  |-  ( ( b  e.  om  /\  ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  e.  S )  ->  ( C `  b )  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
) )
7268, 57, 71syl2anc 644 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( C `  b )  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
) )
7367, 72eleq12d 2506 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( C `  a
)  e.  ( C `
 b )  <->  ( iota_ j  e.  S ( j  i^i  S )  ~~  a )  e.  (
iota_ j  e.  S
( j  i^i  S
)  ~~  b )
) )
7462, 73sylibrd 227 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
( C `  a
)  e.  ( C `
 b ) ) )
75 epel 4500 . . . 4  |-  ( a  _E  b  <->  a  e.  b )
76 fvex 5745 . . . . 5  |-  ( C `
 b )  e. 
_V
7776epelc 4499 . . . 4  |-  ( ( C `  a )  _E  ( C `  b )  <->  ( C `  a )  e.  ( C `  b ) )
7874, 75, 773imtr4g 263 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  _E  b  -> 
( C `  a
)  _E  ( C `
 b ) ) )
7978ralrimivva 2800 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  A. a  e.  om  A. b  e.  om  (
a  _E  b  -> 
( C `  a
)  _E  ( C `
 b ) ) )
80 soisoi 6051 . 2  |-  ( ( (  _E  Or  om  /\  _E  Po  S )  /\  ( C : om -onto-> S  /\  A. a  e.  om  A. b  e. 
om  ( a  _E  b  ->  ( C `  a )  _E  ( C `  b )
) ) )  ->  C  Isom  _E  ,  _E  ( om ,  S ) )
815, 10, 14, 79, 80syl22anc 1186 1  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C  Isom  _E  ,  _E  ( om ,  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E!wreu 2709    i^i cin 3321    C_ wss 3322   class class class wbr 4215    e. cmpt 4269    _E cep 4495    Po wpo 4504    Or wor 4505    We wwe 4543   Ord word 4583   Oncon0 4584   omcom 4848   -onto->wfo 5455   -1-1-onto->wf1o 5456   ` cfv 5457    Isom wiso 5458   iota_crio 6545    ~~ cen 7109    ~< csdm 7111   Fincfn 7112
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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