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Theorem isf33lem 7992
Description: Lemma for isfin3-3 7994. (Contributed by Stefan O'Rear, 17-May-2015.)
Assertion
Ref Expression
isf33lem  |- FinIII  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
Distinct variable group:    g, a, x

Proof of Theorem isf33lem
Dummy variables  b 
f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfin32i 7991 . . . 4  |-  ( f  e. FinIII  ->  -.  om  ~<_*  f )
2 fveq1 5524 . . . . . . . . . . 11  |-  ( a  =  b  ->  (
a `  suc  x )  =  ( b `  suc  x ) )
3 fveq1 5524 . . . . . . . . . . 11  |-  ( a  =  b  ->  (
a `  x )  =  ( b `  x ) )
42, 3sseq12d 3207 . . . . . . . . . 10  |-  ( a  =  b  ->  (
( a `  suc  x )  C_  (
a `  x )  <->  ( b `  suc  x
)  C_  ( b `  x ) ) )
54ralbidv 2563 . . . . . . . . 9  |-  ( a  =  b  ->  ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  <->  A. x  e.  om  (
b `  suc  x ) 
C_  ( b `  x ) ) )
6 rneq 4904 . . . . . . . . . . 11  |-  ( a  =  b  ->  ran  a  =  ran  b )
76inteqd 3867 . . . . . . . . . 10  |-  ( a  =  b  ->  |^| ran  a  =  |^| ran  b
)
87, 6eleq12d 2351 . . . . . . . . 9  |-  ( a  =  b  ->  ( |^| ran  a  e.  ran  a 
<-> 
|^| ran  b  e.  ran  b ) )
95, 8imbi12d 311 . . . . . . . 8  |-  ( a  =  b  ->  (
( A. x  e. 
om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a )  <->  ( A. x  e.  om  (
b `  suc  x ) 
C_  ( b `  x )  ->  |^| ran  b  e.  ran  b ) ) )
109cbvralv 2764 . . . . . . 7  |-  ( A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a )  <->  A. b  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( b `  suc  x )  C_  (
b `  x )  ->  |^| ran  b  e. 
ran  b ) )
11 pweq 3628 . . . . . . . . 9  |-  ( g  =  y  ->  ~P g  =  ~P y
)
1211oveq1d 5873 . . . . . . . 8  |-  ( g  =  y  ->  ( ~P g  ^m  om )  =  ( ~P y  ^m  om ) )
1312raleqdv 2742 . . . . . . 7  |-  ( g  =  y  ->  ( A. b  e.  ( ~P g  ^m  om )
( A. x  e. 
om  ( b `  suc  x )  C_  (
b `  x )  ->  |^| ran  b  e. 
ran  b )  <->  A. b  e.  ( ~P y  ^m  om ) ( A. x  e.  om  ( b `  suc  x )  C_  (
b `  x )  ->  |^| ran  b  e. 
ran  b ) ) )
1410, 13syl5bb 248 . . . . . 6  |-  ( g  =  y  ->  ( A. a  e.  ( ~P g  ^m  om )
( A. x  e. 
om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a )  <->  A. b  e.  ( ~P y  ^m  om ) ( A. x  e.  om  ( b `  suc  x )  C_  (
b `  x )  ->  |^| ran  b  e. 
ran  b ) ) )
1514cbvabv 2402 . . . . 5  |-  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }  =  { y  | 
A. b  e.  ( ~P y  ^m  om ) ( A. x  e.  om  ( b `  suc  x )  C_  (
b `  x )  ->  |^| ran  b  e. 
ran  b ) }
1615isf32lem12 7990 . . . 4  |-  ( f  e. FinIII  ->  ( -.  om  ~<_*  f  ->  f  e.  {
g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) } ) )
171, 16mpd 14 . . 3  |-  ( f  e. FinIII  ->  f  e.  {
g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) } )
1810abbii 2395 . . . 4  |-  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }  =  { g  | 
A. b  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( b `  suc  x )  C_  (
b `  x )  ->  |^| ran  b  e. 
ran  b ) }
1918fin23lem41 7978 . . 3  |-  ( f  e.  { g  | 
A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }  ->  f  e. FinIII )
2017, 19impbii 180 . 2  |-  ( f  e. FinIII  <-> 
f  e.  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) } )
2120eqriv 2280 1  |- FinIII  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543    C_ wss 3152   ~Pcpw 3625   |^|cint 3862   class class class wbr 4023   suc csuc 4394   omcom 4656   ran crn 4690   ` cfv 5255  (class class class)co 5858    ^m cmap 6772    ~<_* cwdom 7271  FinIIIcfin3 7907
This theorem is referenced by:  isfin3-2  7993  isfin3-3  7994  fin23  8015
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-wdom 7273  df-card 7572  df-fin4 7913  df-fin3 7914
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