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Theorem isf33lem 8556
Description: Lemma for isfin3-3 8558. (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 8555 . . . 4  |-  ( f  e. FinIII  ->  -.  om  ~<_*  f )
2 fveq1 5711 . . . . . . . . . . 11  |-  ( a  =  b  ->  (
a `  suc  x )  =  ( b `  suc  x ) )
3 fveq1 5711 . . . . . . . . . . 11  |-  ( a  =  b  ->  (
a `  x )  =  ( b `  x ) )
42, 3sseq12d 3406 . . . . . . . . . 10  |-  ( a  =  b  ->  (
( a `  suc  x )  C_  (
a `  x )  <->  ( b `  suc  x
)  C_  ( b `  x ) ) )
54ralbidv 2756 . . . . . . . . 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 5086 . . . . . . . . . . 11  |-  ( a  =  b  ->  ran  a  =  ran  b )
76inteqd 4154 . . . . . . . . . 10  |-  ( a  =  b  ->  |^| ran  a  =  |^| ran  b
)
87, 6eleq12d 2511 . . . . . . . . 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 2968 . . . . . . 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 3884 . . . . . . . . 9  |-  ( g  =  y  ->  ~P g  =  ~P y
)
1211oveq1d 6127 . . . . . . . 8  |-  ( g  =  y  ->  ( ~P g  ^m  om )  =  ( ~P y  ^m  om ) )
1312raleqdv 2944 . . . . . . 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 2570 . . . . 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 8554 . . . 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 2561 . . . 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 8542 . . 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 2440 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 1369    e. wcel 1756   {cab 2429   A.wral 2736    C_ wss 3349   ~Pcpw 3881   |^|cint 4149   class class class wbr 4313   suc csuc 4742   ran crn 4862   ` cfv 5439  (class class class)co 6112   omcom 6497    ^m cmap 7235    ~<_* cwdom 7793  FinIIIcfin3 8471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-seqom 6924  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-wdom 7795  df-card 8130  df-fin4 8477  df-fin3 8478
This theorem is referenced by:  isfin3-2  8557  isfin3-3  8558  fin23  8579
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