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Theorem isfin3ds 8498
Description: Property of a III-finite set (descending sequence version). (Contributed by Mario Carneiro, 16-May-2015.)
Hypothesis
Ref Expression
isfin3ds.f  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. b  e.  om  ( a `  suc  b )  C_  (
a `  b )  ->  |^| ran  a  e. 
ran  a ) }
Assertion
Ref Expression
isfin3ds  |-  ( A  e.  V  ->  ( A  e.  F  <->  A. f  e.  ( ~P A  ^m  om ) ( A. x  e.  om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) ) )
Distinct variable group:    a, b, f, g, x, A
Allowed substitution hints:    F( x, f, g, a, b)    V( x, f, g, a, b)

Proof of Theorem isfin3ds
StepHypRef Expression
1 suceq 4784 . . . . . . . . 9  |-  ( b  =  x  ->  suc  b  =  suc  x )
21fveq2d 5695 . . . . . . . 8  |-  ( b  =  x  ->  (
a `  suc  b )  =  ( a `  suc  x ) )
3 fveq2 5691 . . . . . . . 8  |-  ( b  =  x  ->  (
a `  b )  =  ( a `  x ) )
42, 3sseq12d 3385 . . . . . . 7  |-  ( b  =  x  ->  (
( a `  suc  b )  C_  (
a `  b )  <->  ( a `  suc  x
)  C_  ( a `  x ) ) )
54cbvralv 2947 . . . . . 6  |-  ( A. b  e.  om  (
a `  suc  b ) 
C_  ( a `  b )  <->  A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )
)
6 fveq1 5690 . . . . . . . 8  |-  ( a  =  f  ->  (
a `  suc  x )  =  ( f `  suc  x ) )
7 fveq1 5690 . . . . . . . 8  |-  ( a  =  f  ->  (
a `  x )  =  ( f `  x ) )
86, 7sseq12d 3385 . . . . . . 7  |-  ( a  =  f  ->  (
( a `  suc  x )  C_  (
a `  x )  <->  ( f `  suc  x
)  C_  ( f `  x ) ) )
98ralbidv 2735 . . . . . 6  |-  ( a  =  f  ->  ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  <->  A. x  e.  om  (
f `  suc  x ) 
C_  ( f `  x ) ) )
105, 9syl5bb 257 . . . . 5  |-  ( a  =  f  ->  ( A. b  e.  om  ( a `  suc  b )  C_  (
a `  b )  <->  A. x  e.  om  (
f `  suc  x ) 
C_  ( f `  x ) ) )
11 rneq 5065 . . . . . . 7  |-  ( a  =  f  ->  ran  a  =  ran  f )
1211inteqd 4133 . . . . . 6  |-  ( a  =  f  ->  |^| ran  a  =  |^| ran  f
)
1312, 11eleq12d 2511 . . . . 5  |-  ( a  =  f  ->  ( |^| ran  a  e.  ran  a 
<-> 
|^| ran  f  e.  ran  f ) )
1410, 13imbi12d 320 . . . 4  |-  ( a  =  f  ->  (
( A. b  e. 
om  ( a `  suc  b )  C_  (
a `  b )  ->  |^| ran  a  e. 
ran  a )  <->  ( A. x  e.  om  (
f `  suc  x ) 
C_  ( f `  x )  ->  |^| ran  f  e.  ran  f ) ) )
1514cbvralv 2947 . . 3  |-  ( A. a  e.  ( ~P g  ^m  om ) ( A. b  e.  om  ( a `  suc  b )  C_  (
a `  b )  ->  |^| ran  a  e. 
ran  a )  <->  A. f  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) )
16 pweq 3863 . . . . 5  |-  ( g  =  A  ->  ~P g  =  ~P A
)
1716oveq1d 6106 . . . 4  |-  ( g  =  A  ->  ( ~P g  ^m  om )  =  ( ~P A  ^m  om ) )
1817raleqdv 2923 . . 3  |-  ( g  =  A  ->  ( A. f  e.  ( ~P g  ^m  om )
( A. x  e. 
om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f )  <->  A. f  e.  ( ~P A  ^m  om ) ( A. x  e.  om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) ) )
1915, 18syl5bb 257 . 2  |-  ( g  =  A  ->  ( A. a  e.  ( ~P g  ^m  om )
( A. b  e. 
om  ( a `  suc  b )  C_  (
a `  b )  ->  |^| ran  a  e. 
ran  a )  <->  A. f  e.  ( ~P A  ^m  om ) ( A. x  e.  om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) ) )
20 isfin3ds.f . 2  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. b  e.  om  ( a `  suc  b )  C_  (
a `  b )  ->  |^| ran  a  e. 
ran  a ) }
2119, 20elab2g 3108 1  |-  ( A  e.  V  ->  ( A  e.  F  <->  A. f  e.  ( ~P A  ^m  om ) ( A. x  e.  om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2715    C_ wss 3328   ~Pcpw 3860   |^|cint 4128   suc csuc 4721   ran crn 4841   ` cfv 5418  (class class class)co 6091   omcom 6476    ^m cmap 7214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-suc 4725  df-cnv 4848  df-dm 4850  df-rn 4851  df-iota 5381  df-fv 5426  df-ov 6094
This theorem is referenced by:  ssfin3ds  8499  fin23lem17  8507  fin23lem39  8519  fin23lem40  8520  isf32lem12  8533  isfin3-3  8537
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