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Theorem isfin3ds 7955
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 4457 . . . . . . . . 9  |-  ( b  =  x  ->  suc  b  =  suc  x )
21fveq2d 5529 . . . . . . . 8  |-  ( b  =  x  ->  (
a `  suc  b )  =  ( a `  suc  x ) )
3 fveq2 5525 . . . . . . . 8  |-  ( b  =  x  ->  (
a `  b )  =  ( a `  x ) )
42, 3sseq12d 3207 . . . . . . 7  |-  ( b  =  x  ->  (
( a `  suc  b )  C_  (
a `  b )  <->  ( a `  suc  x
)  C_  ( a `  x ) ) )
54cbvralv 2764 . . . . . 6  |-  ( A. b  e.  om  (
a `  suc  b ) 
C_  ( a `  b )  <->  A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )
)
6 fveq1 5524 . . . . . . . 8  |-  ( a  =  f  ->  (
a `  suc  x )  =  ( f `  suc  x ) )
7 fveq1 5524 . . . . . . . 8  |-  ( a  =  f  ->  (
a `  x )  =  ( f `  x ) )
86, 7sseq12d 3207 . . . . . . 7  |-  ( a  =  f  ->  (
( a `  suc  x )  C_  (
a `  x )  <->  ( f `  suc  x
)  C_  ( f `  x ) ) )
98ralbidv 2563 . . . . . 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 248 . . . . 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 4904 . . . . . . 7  |-  ( a  =  f  ->  ran  a  =  ran  f )
1211inteqd 3867 . . . . . 6  |-  ( a  =  f  ->  |^| ran  a  =  |^| ran  f
)
1312, 11eleq12d 2351 . . . . 5  |-  ( a  =  f  ->  ( |^| ran  a  e.  ran  a 
<-> 
|^| ran  f  e.  ran  f ) )
1410, 13imbi12d 311 . . . 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 2764 . . 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 3628 . . . . 5  |-  ( g  =  A  ->  ~P g  =  ~P A
)
1716oveq1d 5873 . . . 4  |-  ( g  =  A  ->  ( ~P g  ^m  om )  =  ( ~P A  ^m  om ) )
1817raleqdv 2742 . . 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 248 . 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 2916 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 set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543    C_ wss 3152   ~Pcpw 3625   |^|cint 3862   suc csuc 4394   omcom 4656   ran crn 4690   ` cfv 5255  (class class class)co 5858    ^m cmap 6772
This theorem is referenced by:  ssfin3ds  7956  fin23lem17  7964  fin23lem39  7976  fin23lem40  7977  isf32lem12  7990  isfin3-3  7994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-suc 4398  df-cnv 4697  df-dm 4699  df-rn 4700  df-iota 5219  df-fv 5263  df-ov 5861
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