MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfin3ds Structured version   Visualization version   Unicode version

Theorem isfin3ds 8777
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 5495 . . . . . . . . 9  |-  ( b  =  x  ->  suc  b  =  suc  x )
21fveq2d 5883 . . . . . . . 8  |-  ( b  =  x  ->  (
a `  suc  b )  =  ( a `  suc  x ) )
3 fveq2 5879 . . . . . . . 8  |-  ( b  =  x  ->  (
a `  b )  =  ( a `  x ) )
42, 3sseq12d 3447 . . . . . . 7  |-  ( b  =  x  ->  (
( a `  suc  b )  C_  (
a `  b )  <->  ( a `  suc  x
)  C_  ( a `  x ) ) )
54cbvralv 3005 . . . . . 6  |-  ( A. b  e.  om  (
a `  suc  b ) 
C_  ( a `  b )  <->  A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )
)
6 fveq1 5878 . . . . . . . 8  |-  ( a  =  f  ->  (
a `  suc  x )  =  ( f `  suc  x ) )
7 fveq1 5878 . . . . . . . 8  |-  ( a  =  f  ->  (
a `  x )  =  ( f `  x ) )
86, 7sseq12d 3447 . . . . . . 7  |-  ( a  =  f  ->  (
( a `  suc  x )  C_  (
a `  x )  <->  ( f `  suc  x
)  C_  ( f `  x ) ) )
98ralbidv 2829 . . . . . 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 265 . . . . 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 5066 . . . . . . 7  |-  ( a  =  f  ->  ran  a  =  ran  f )
1211inteqd 4231 . . . . . 6  |-  ( a  =  f  ->  |^| ran  a  =  |^| ran  f
)
1312, 11eleq12d 2543 . . . . 5  |-  ( a  =  f  ->  ( |^| ran  a  e.  ran  a 
<-> 
|^| ran  f  e.  ran  f ) )
1410, 13imbi12d 327 . . . 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 3005 . . 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 3945 . . . . 5  |-  ( g  =  A  ->  ~P g  =  ~P A
)
1716oveq1d 6323 . . . 4  |-  ( g  =  A  ->  ( ~P g  ^m  om )  =  ( ~P A  ^m  om ) )
1817raleqdv 2979 . . 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 265 . 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 3175 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 189    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756    C_ wss 3390   ~Pcpw 3942   |^|cint 4226   ran crn 4840   suc csuc 5432   ` cfv 5589  (class class class)co 6308   omcom 6711    ^m cmap 7490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-cnv 4847  df-dm 4849  df-rn 4850  df-suc 5436  df-iota 5553  df-fv 5597  df-ov 6311
This theorem is referenced by:  ssfin3ds  8778  fin23lem17  8786  fin23lem39  8798  fin23lem40  8799  isf32lem12  8812  isfin3-3  8816
  Copyright terms: Public domain W3C validator