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Theorem isfin3ds 8042
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 4536 . . . . . . . . 9  |-  ( b  =  x  ->  suc  b  =  suc  x )
21fveq2d 5609 . . . . . . . 8  |-  ( b  =  x  ->  (
a `  suc  b )  =  ( a `  suc  x ) )
3 fveq2 5605 . . . . . . . 8  |-  ( b  =  x  ->  (
a `  b )  =  ( a `  x ) )
42, 3sseq12d 3283 . . . . . . 7  |-  ( b  =  x  ->  (
( a `  suc  b )  C_  (
a `  b )  <->  ( a `  suc  x
)  C_  ( a `  x ) ) )
54cbvralv 2840 . . . . . 6  |-  ( A. b  e.  om  (
a `  suc  b ) 
C_  ( a `  b )  <->  A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )
)
6 fveq1 5604 . . . . . . . 8  |-  ( a  =  f  ->  (
a `  suc  x )  =  ( f `  suc  x ) )
7 fveq1 5604 . . . . . . . 8  |-  ( a  =  f  ->  (
a `  x )  =  ( f `  x ) )
86, 7sseq12d 3283 . . . . . . 7  |-  ( a  =  f  ->  (
( a `  suc  x )  C_  (
a `  x )  <->  ( f `  suc  x
)  C_  ( f `  x ) ) )
98ralbidv 2639 . . . . . 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 4983 . . . . . . 7  |-  ( a  =  f  ->  ran  a  =  ran  f )
1211inteqd 3946 . . . . . 6  |-  ( a  =  f  ->  |^| ran  a  =  |^| ran  f
)
1312, 11eleq12d 2426 . . . . 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 2840 . . 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 3704 . . . . 5  |-  ( g  =  A  ->  ~P g  =  ~P A
)
1716oveq1d 5957 . . . 4  |-  ( g  =  A  ->  ( ~P g  ^m  om )  =  ( ~P A  ^m  om ) )
1817raleqdv 2818 . . 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 2992 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 1642    e. wcel 1710   {cab 2344   A.wral 2619    C_ wss 3228   ~Pcpw 3701   |^|cint 3941   suc csuc 4473   omcom 4735   ran crn 4769   ` cfv 5334  (class class class)co 5942    ^m cmap 6857
This theorem is referenced by:  ssfin3ds  8043  fin23lem17  8051  fin23lem39  8063  fin23lem40  8064  isf32lem12  8077  isfin3-3  8081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-int 3942  df-br 4103  df-opab 4157  df-suc 4477  df-cnv 4776  df-dm 4778  df-rn 4779  df-iota 5298  df-fv 5342  df-ov 5945
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