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Theorem isf33lem 8778
Description: Lemma for isfin3-3 8780. (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 8777 . . . 4  |-  ( f  e. FinIII  ->  -.  om  ~<_*  f )
2 fveq1 5848 . . . . . . . . . . 11  |-  ( a  =  b  ->  (
a `  suc  x )  =  ( b `  suc  x ) )
3 fveq1 5848 . . . . . . . . . . 11  |-  ( a  =  b  ->  (
a `  x )  =  ( b `  x ) )
42, 3sseq12d 3471 . . . . . . . . . 10  |-  ( a  =  b  ->  (
( a `  suc  x )  C_  (
a `  x )  <->  ( b `  suc  x
)  C_  ( b `  x ) ) )
54ralbidv 2843 . . . . . . . . 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 5049 . . . . . . . . . . 11  |-  ( a  =  b  ->  ran  a  =  ran  b )
76inteqd 4232 . . . . . . . . . 10  |-  ( a  =  b  ->  |^| ran  a  =  |^| ran  b
)
87, 6eleq12d 2484 . . . . . . . . 9  |-  ( a  =  b  ->  ( |^| ran  a  e.  ran  a 
<-> 
|^| ran  b  e.  ran  b ) )
95, 8imbi12d 318 . . . . . . . 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 3034 . . . . . . 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 3958 . . . . . . . . 9  |-  ( g  =  y  ->  ~P g  =  ~P y
)
1211oveq1d 6293 . . . . . . . 8  |-  ( g  =  y  ->  ( ~P g  ^m  om )  =  ( ~P y  ^m  om ) )
1312raleqdv 3010 . . . . . . 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 2545 . . . . 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 8776 . . . 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 2536 . . . 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 8764 . . 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 187 . 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 2398 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 1405    e. wcel 1842   {cab 2387   A.wral 2754    C_ wss 3414   ~Pcpw 3955   |^|cint 4227   class class class wbr 4395   ran crn 4824   suc csuc 5412   ` cfv 5569  (class class class)co 6278   omcom 6683    ^m cmap 7457    ~<_* cwdom 8017  FinIIIcfin3 8693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-seqom 7150  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-wdom 8019  df-card 8352  df-fin4 8699  df-fin3 8700
This theorem is referenced by:  isfin3-2  8779  isfin3-3  8780  fin23  8801
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