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Theorem isfin3ds 8173
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 4614 . . . . . . . . 9  |-  ( b  =  x  ->  suc  b  =  suc  x )
21fveq2d 5699 . . . . . . . 8  |-  ( b  =  x  ->  (
a `  suc  b )  =  ( a `  suc  x ) )
3 fveq2 5695 . . . . . . . 8  |-  ( b  =  x  ->  (
a `  b )  =  ( a `  x ) )
42, 3sseq12d 3345 . . . . . . 7  |-  ( b  =  x  ->  (
( a `  suc  b )  C_  (
a `  b )  <->  ( a `  suc  x
)  C_  ( a `  x ) ) )
54cbvralv 2900 . . . . . 6  |-  ( A. b  e.  om  (
a `  suc  b ) 
C_  ( a `  b )  <->  A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )
)
6 fveq1 5694 . . . . . . . 8  |-  ( a  =  f  ->  (
a `  suc  x )  =  ( f `  suc  x ) )
7 fveq1 5694 . . . . . . . 8  |-  ( a  =  f  ->  (
a `  x )  =  ( f `  x ) )
86, 7sseq12d 3345 . . . . . . 7  |-  ( a  =  f  ->  (
( a `  suc  x )  C_  (
a `  x )  <->  ( f `  suc  x
)  C_  ( f `  x ) ) )
98ralbidv 2694 . . . . . 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 249 . . . . 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 5062 . . . . . . 7  |-  ( a  =  f  ->  ran  a  =  ran  f )
1211inteqd 4023 . . . . . 6  |-  ( a  =  f  ->  |^| ran  a  =  |^| ran  f
)
1312, 11eleq12d 2480 . . . . 5  |-  ( a  =  f  ->  ( |^| ran  a  e.  ran  a 
<-> 
|^| ran  f  e.  ran  f ) )
1410, 13imbi12d 312 . . . 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 2900 . . 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 3770 . . . . 5  |-  ( g  =  A  ->  ~P g  =  ~P A
)
1716oveq1d 6063 . . . 4  |-  ( g  =  A  ->  ( ~P g  ^m  om )  =  ( ~P A  ^m  om ) )
1817raleqdv 2878 . . 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 249 . 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 3052 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 177    = wceq 1649    e. wcel 1721   {cab 2398   A.wral 2674    C_ wss 3288   ~Pcpw 3767   |^|cint 4018   suc csuc 4551   omcom 4812   ran crn 4846   ` cfv 5421  (class class class)co 6048    ^m cmap 6985
This theorem is referenced by:  ssfin3ds  8174  fin23lem17  8182  fin23lem39  8194  fin23lem40  8195  isf32lem12  8208  isfin3-3  8212
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-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-int 4019  df-br 4181  df-opab 4235  df-suc 4555  df-cnv 4853  df-dm 4855  df-rn 4856  df-iota 5385  df-fv 5429  df-ov 6051
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