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Theorem fin23lem40 7993
Description: Lemma for fin23 8031. FinII sets satisfy the descending chain condition. (Contributed by Stefan O'Rear, 3-Nov-2014.)
Hypothesis
Ref Expression
fin23lem40.f  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
Assertion
Ref Expression
fin23lem40  |-  ( A  e. FinII  ->  A  e.  F
)
Distinct variable groups:    g, a, x, A    F, a
Allowed substitution hints:    F( x, g)

Proof of Theorem fin23lem40
Dummy variables  b 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmapi 6808 . . . 4  |-  ( f  e.  ( ~P A  ^m  om )  ->  f : om --> ~P A )
2 simpl 443 . . . . . 6  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  A  e. FinII )
3 frn 5411 . . . . . . 7  |-  ( f : om --> ~P A  ->  ran  f  C_  ~P A )
43ad2antrl 708 . . . . . 6  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  ran  f  C_  ~P A )
5 fdm 5409 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  dom  f  =  om )
6 peano1 4691 . . . . . . . . . 10  |-  (/)  e.  om
7 ne0i 3474 . . . . . . . . . 10  |-  ( (/)  e.  om  ->  om  =/=  (/) )
86, 7mp1i 11 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  om  =/=  (/) )
95, 8eqnetrd 2477 . . . . . . . 8  |-  ( f : om --> ~P A  ->  dom  f  =/=  (/) )
10 dm0rn0 4911 . . . . . . . . 9  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
1110necon3bii 2491 . . . . . . . 8  |-  ( dom  f  =/=  (/)  <->  ran  f  =/=  (/) )
129, 11sylib 188 . . . . . . 7  |-  ( f : om --> ~P A  ->  ran  f  =/=  (/) )
1312ad2antrl 708 . . . . . 6  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  ran  f  =/=  (/) )
14 ffn 5405 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  f  Fn  om )
1514ad2antrl 708 . . . . . . . 8  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  f  Fn  om )
16 sspss 3288 . . . . . . . . . . 11  |-  ( ( f `  suc  b
)  C_  ( f `  b )  <->  ( (
f `  suc  b ) 
C.  ( f `  b )  \/  (
f `  suc  b )  =  ( f `  b ) ) )
17 fvex 5555 . . . . . . . . . . . . . . 15  |-  ( f `
 b )  e. 
_V
18 fvex 5555 . . . . . . . . . . . . . . 15  |-  ( f `
 suc  b )  e.  _V
1917, 18brcnv 4880 . . . . . . . . . . . . . 14  |-  ( ( f `  b ) `' [ C.]  ( f `  suc  b )  <->  ( f `  suc  b ) [ C.]  ( f `  b
) )
2017brrpss 6296 . . . . . . . . . . . . . 14  |-  ( ( f `  suc  b
) [ C.]  ( f `  b )  <->  ( f `  suc  b )  C.  ( f `  b
) )
2119, 20bitri 240 . . . . . . . . . . . . 13  |-  ( ( f `  b ) `' [ C.]  ( f `  suc  b )  <->  ( f `  suc  b )  C.  ( f `  b
) )
22 eqcom 2298 . . . . . . . . . . . . 13  |-  ( ( f `  b )  =  ( f `  suc  b )  <->  ( f `  suc  b )  =  ( f `  b
) )
2321, 22orbi12i 507 . . . . . . . . . . . 12  |-  ( ( ( f `  b
) `' [ C.]  (
f `  suc  b )  \/  ( f `  b )  =  ( f `  suc  b
) )  <->  ( (
f `  suc  b ) 
C.  ( f `  b )  \/  (
f `  suc  b )  =  ( f `  b ) ) )
2423biimpri 197 . . . . . . . . . . 11  |-  ( ( ( f `  suc  b )  C.  (
f `  b )  \/  ( f `  suc  b )  =  ( f `  b ) )  ->  ( (
f `  b ) `' [ C.]  ( f `  suc  b )  \/  (
f `  b )  =  ( f `  suc  b ) ) )
2516, 24sylbi 187 . . . . . . . . . 10  |-  ( ( f `  suc  b
)  C_  ( f `  b )  ->  (
( f `  b
) `' [ C.]  (
f `  suc  b )  \/  ( f `  b )  =  ( f `  suc  b
) ) )
2625ralimi 2631 . . . . . . . . 9  |-  ( A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b )  ->  A. b  e.  om  ( ( f `
 b ) `' [
C.]  ( f `  suc  b )  \/  (
f `  b )  =  ( f `  suc  b ) ) )
2726ad2antll 709 . . . . . . . 8  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  A. b  e.  om  ( ( f `  b ) `' [ C.]  ( f `  suc  b )  \/  (
f `  b )  =  ( f `  suc  b ) ) )
28 porpss 6297 . . . . . . . . . 10  |- [ C.]  Po  ran  f
29 cnvpo 5229 . . . . . . . . . 10  |-  ( [ C.]  Po  ran  f  <->  `' [ C.]  Po  ran  f )
3028, 29mpbi 199 . . . . . . . . 9  |-  `' [ C.]  Po  ran  f
3130a1i 10 . . . . . . . 8  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  `' [ C.]  Po  ran  f )
32 sornom 7919 . . . . . . . 8  |-  ( ( f  Fn  om  /\  A. b  e.  om  (
( f `  b
) `' [ C.]  (
f `  suc  b )  \/  ( f `  b )  =  ( f `  suc  b
) )  /\  `' [ C.] 
Po  ran  f )  ->  `' [ C.]  Or  ran  f
)
3315, 27, 31, 32syl3anc 1182 . . . . . . 7  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  `' [ C.]  Or  ran  f )
34 cnvso 5230 . . . . . . 7  |-  ( [ C.]  Or  ran  f  <->  `' [ C.]  Or  ran  f )
3533, 34sylibr 203 . . . . . 6  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  -> [ C.]  Or  ran  f )
36 fin2i2 7960 . . . . . 6  |-  ( ( ( A  e. FinII  /\  ran  f  C_  ~P A )  /\  ( ran  f  =/=  (/)  /\ [ C.]  Or  ran  f ) )  ->  |^| ran  f  e.  ran  f )
372, 4, 13, 35, 36syl22anc 1183 . . . . 5  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  |^| ran  f  e. 
ran  f )
3837expr 598 . . . 4  |-  ( ( A  e. FinII  /\  f : om
--> ~P A )  -> 
( A. b  e. 
om  ( f `  suc  b )  C_  (
f `  b )  ->  |^| ran  f  e. 
ran  f ) )
391, 38sylan2 460 . . 3  |-  ( ( A  e. FinII  /\  f  e.  ( ~P A  ^m  om ) )  ->  ( A. b  e.  om  ( f `  suc  b )  C_  (
f `  b )  ->  |^| ran  f  e. 
ran  f ) )
4039ralrimiva 2639 . 2  |-  ( A  e. FinII  ->  A. f  e.  ( ~P A  ^m  om ) ( A. b  e.  om  ( f `  suc  b )  C_  (
f `  b )  ->  |^| ran  f  e. 
ran  f ) )
41 fin23lem40.f . . 3  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
4241isfin3ds 7971 . 2  |-  ( A  e. FinII  ->  ( A  e.  F  <->  A. f  e.  ( ~P A  ^m  om ) ( A. b  e.  om  ( f `  suc  b )  C_  (
f `  b )  ->  |^| ran  f  e. 
ran  f ) ) )
4340, 42mpbird 223 1  |-  ( A  e. FinII  ->  A  e.  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556    C_ wss 3165    C. wpss 3166   (/)c0 3468   ~Pcpw 3638   |^|cint 3878   class class class wbr 4039    Po wpo 4328    Or wor 4329   suc csuc 4410   omcom 4672   `'ccnv 4704   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   [ C.] crpss 6292    ^m cmap 6788  FinIIcfin2 7921
This theorem is referenced by:  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-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-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-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-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-rpss 6293  df-map 6790  df-fin2 7928
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