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Theorem fin23lem27 8168
Description: The mapping constructed in fin23lem22 8167 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 4817 . . . 4  |-  Ord  om
2 ordwe 4558 . . . 4  |-  ( Ord 
om  ->  _E  We  om )
3 weso 4537 . . . 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 4484 . . . . 5  |-  (  _E  Or  om  ->  _E  Po  om )
74, 6ax-mp 8 . . . 4  |-  _E  Po  om
8 poss 4469 . . . 4  |-  ( S 
C_  om  ->  (  _E  Po  om  ->  _E  Po  S ) )
97, 8mpi 17 . . 3  |-  ( S 
C_  om  ->  _E  Po  S )
109adantr 452 . 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 8167 . . 3  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -1-1-onto-> S )
13 f1ofo 5644 . . 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 7837 . . . . . . . 8  |-  ( ( a  e.  om  /\  b  e.  om )  ->  ( a  e.  b  <-> 
a  ~<  b ) )
1615adantl 453 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  <->  a  ~<  b ) )
1716biimpd 199 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
a  ~<  b ) )
18 fin23lem23 8166 . . . . . . . . . . . . 13  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  a
)
1918adantrr 698 . . . . . . . . . . . 12  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! j  e.  S  (
j  i^i  S )  ~~  a )
20 ineq1 3499 . . . . . . . . . . . . . 14  |-  ( j  =  i  ->  (
j  i^i  S )  =  ( i  i^i 
S ) )
2120breq1d 4186 . . . . . . . . . . . . 13  |-  ( j  =  i  ->  (
( j  i^i  S
)  ~~  a  <->  ( i  i^i  S )  ~~  a
) )
2221cbvreuv 2898 . . . . . . . . . . . 12  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  a  <->  E! i  e.  S  ( i  i^i  S )  ~~  a
)
2319, 22sylib 189 . . . . . . . . . . 11  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! i  e.  S  (
i  i^i  S )  ~~  a )
24 nfv 1626 . . . . . . . . . . . 12  |-  F/ i ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~~  a
2521cbvriotav 6524 . . . . . . . . . . . 12  |-  ( iota_ j  e.  S ( j  i^i  S )  ~~  a )  =  (
iota_ i  e.  S
( i  i^i  S
)  ~~  a )
26 ineq1 3499 . . . . . . . . . . . . 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 4186 . . . . . . . . . . . 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 6536 . . . . . . . . . . 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 450 . . . . . . . . 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 698 . . . . . . . 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 734 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
a  ~<  b )
33 fin23lem23 8166 . . . . . . . . . . . . . . 15  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  b  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  b
)
3433adantrl 697 . . . . . . . . . . . . . 14  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! j  e.  S  (
j  i^i  S )  ~~  b )
3520breq1d 4186 . . . . . . . . . . . . . . 15  |-  ( j  =  i  ->  (
( j  i^i  S
)  ~~  b  <->  ( i  i^i  S )  ~~  b
) )
3635cbvreuv 2898 . . . . . . . . . . . . . 14  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  b  <->  E! i  e.  S  ( i  i^i  S )  ~~  b
)
3734, 36sylib 189 . . . . . . . . . . . . 13  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! i  e.  S  (
i  i^i  S )  ~~  b )
38 nfv 1626 . . . . . . . . . . . . . 14  |-  F/ i ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )  ~~  b
3935cbvriotav 6524 . . . . . . . . . . . . . 14  |-  ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  =  (
iota_ i  e.  S
( i  i^i  S
)  ~~  b )
40 ineq1 3499 . . . . . . . . . . . . . . 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 4186 . . . . . . . . . . . . . 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 6536 . . . . . . . . . . . . 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 450 . . . . . . . . . . 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 7121 . . . . . . . . . 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 698 . . . . . . . . 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 7204 . . . . . . . . 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 643 . . . . . . . 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 7206 . . . . . . . 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 643 . . . . . . 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 599 . . . . . 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 731 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  S  C_ 
om )
53 omsson 4812 . . . . . . . . 9  |-  om  C_  On
5452, 53syl6ss 3324 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  S  C_  On )
5529simpld 446 . . . . . . . 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 3313 . . . . . . 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 446 . . . . . . . 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 3313 . . . . . . 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 7219 . . . . . . . 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 1155 . . . . . . 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 643 . . . . . 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 53 . . . . 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 733 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  a  e.  om )
64 breq2 4180 . . . . . . . . 9  |-  ( i  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( j  i^i  S )  ~~  a
) )
6564riotabidv 6514 . . . . . . . 8  |-  ( i  =  a  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  i )  =  ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )
)
6665, 11fvmptg 5767 . . . . . . 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 643 . . . . . 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 734 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  b  e.  om )
69 breq2 4180 . . . . . . . . 9  |-  ( i  =  b  ->  (
( j  i^i  S
)  ~~  i  <->  ( j  i^i  S )  ~~  b
) )
7069riotabidv 6514 . . . . . . . 8  |-  ( i  =  b  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  i )  =  ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )
)
7170, 11fvmptg 5767 . . . . . . 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 643 . . . . . 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 2476 . . . . 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 226 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
( C `  a
)  e.  ( C `
 b ) ) )
75 epel 4461 . . . 4  |-  ( a  _E  b  <->  a  e.  b )
76 fvex 5705 . . . . 5  |-  ( C `
 b )  e. 
_V
7776epelc 4460 . . . 4  |-  ( ( C `  a )  _E  ( C `  b )  <->  ( C `  a )  e.  ( C `  b ) )
7874, 75, 773imtr4g 262 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  _E  b  -> 
( C `  a
)  _E  ( C `
 b ) ) )
7978ralrimivva 2762 . 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 6011 . 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 1185 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   E!wreu 2672    i^i cin 3283    C_ wss 3284   class class class wbr 4176    e. cmpt 4230    _E cep 4456    Po wpo 4465    Or wor 4466    We wwe 4504   Ord word 4544   Oncon0 4545   omcom 4808   -onto->wfo 5415   -1-1-onto->wf1o 5416   ` cfv 5417    Isom wiso 5418   iota_crio 6505    ~~ cen 7069    ~< csdm 7071   Fincfn 7072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6512  df-recs 6596  df-1o 6687  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-card 7786
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