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Theorem isf33lem 8735
Description: Lemma for isfin3-3 8737. (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 8734 . . . 4  |-  ( f  e. FinIII  ->  -.  om  ~<_*  f )
2 fveq1 5856 . . . . . . . . . . 11  |-  ( a  =  b  ->  (
a `  suc  x )  =  ( b `  suc  x ) )
3 fveq1 5856 . . . . . . . . . . 11  |-  ( a  =  b  ->  (
a `  x )  =  ( b `  x ) )
42, 3sseq12d 3526 . . . . . . . . . 10  |-  ( a  =  b  ->  (
( a `  suc  x )  C_  (
a `  x )  <->  ( b `  suc  x
)  C_  ( b `  x ) ) )
54ralbidv 2896 . . . . . . . . 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 5219 . . . . . . . . . . 11  |-  ( a  =  b  ->  ran  a  =  ran  b )
76inteqd 4280 . . . . . . . . . 10  |-  ( a  =  b  ->  |^| ran  a  =  |^| ran  b
)
87, 6eleq12d 2542 . . . . . . . . 9  |-  ( a  =  b  ->  ( |^| ran  a  e.  ran  a 
<-> 
|^| ran  b  e.  ran  b ) )
95, 8imbi12d 320 . . . . . . . 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 3081 . . . . . . 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 4006 . . . . . . . . 9  |-  ( g  =  y  ->  ~P g  =  ~P y
)
1211oveq1d 6290 . . . . . . . 8  |-  ( g  =  y  ->  ( ~P g  ^m  om )  =  ( ~P y  ^m  om ) )
1312raleqdv 3057 . . . . . . 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 257 . . . . . 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 2603 . . . . 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 8733 . . . 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 15 . . 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 2594 . . . 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 8721 . . 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 188 . 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 2456 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1374    e. wcel 1762   {cab 2445   A.wral 2807    C_ wss 3469   ~Pcpw 4003   |^|cint 4275   class class class wbr 4440   suc csuc 4873   ran crn 4993   ` cfv 5579  (class class class)co 6275   omcom 6671    ^m cmap 7410    ~<_* cwdom 7972  FinIIIcfin3 8650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-seqom 7103  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-wdom 7974  df-card 8309  df-fin4 8656  df-fin3 8657
This theorem is referenced by:  isfin3-2  8736  isfin3-3  8737  fin23  8758
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