MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isf33lem Unicode version

Theorem isf33lem 8008
Description: Lemma for isfin3-3 8010. (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 8007 . . . 4  |-  ( f  e. FinIII  ->  -.  om  ~<_*  f )
2 fveq1 5540 . . . . . . . . . . 11  |-  ( a  =  b  ->  (
a `  suc  x )  =  ( b `  suc  x ) )
3 fveq1 5540 . . . . . . . . . . 11  |-  ( a  =  b  ->  (
a `  x )  =  ( b `  x ) )
42, 3sseq12d 3220 . . . . . . . . . 10  |-  ( a  =  b  ->  (
( a `  suc  x )  C_  (
a `  x )  <->  ( b `  suc  x
)  C_  ( b `  x ) ) )
54ralbidv 2576 . . . . . . . . 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 4920 . . . . . . . . . . 11  |-  ( a  =  b  ->  ran  a  =  ran  b )
76inteqd 3883 . . . . . . . . . 10  |-  ( a  =  b  ->  |^| ran  a  =  |^| ran  b
)
87, 6eleq12d 2364 . . . . . . . . 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 2777 . . . . . . 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 3641 . . . . . . . . 9  |-  ( g  =  y  ->  ~P g  =  ~P y
)
1211oveq1d 5889 . . . . . . . 8  |-  ( g  =  y  ->  ( ~P g  ^m  om )  =  ( ~P y  ^m  om ) )
1312raleqdv 2755 . . . . . . 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 2415 . . . . 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 8006 . . . 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 2408 . . . 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 7994 . . 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 2293 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 1632    e. wcel 1696   {cab 2282   A.wral 2556    C_ wss 3165   ~Pcpw 3638   |^|cint 3878   class class class wbr 4039   suc csuc 4410   omcom 4672   ran crn 4706   ` cfv 5271  (class class class)co 5874    ^m cmap 6788    ~<_* cwdom 7287  FinIIIcfin3 7923
This theorem is referenced by:  isfin3-2  8009  isfin3-3  8010  fin23  8031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-wdom 7289  df-card 7588  df-fin4 7929  df-fin3 7930
  Copyright terms: Public domain W3C validator