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Theorem fin23lem40 8743
Description: Lemma for fin23 8781. 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 7452 . . . 4  |-  ( f  e.  ( ~P A  ^m  om )  ->  f : om --> ~P A )
2 simpl 457 . . . . . 6  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  A  e. FinII )
3 frn 5743 . . . . . . 7  |-  ( f : om --> ~P A  ->  ran  f  C_  ~P A )
43ad2antrl 727 . . . . . 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 5741 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  dom  f  =  om )
6 peano1 6714 . . . . . . . . . 10  |-  (/)  e.  om
7 ne0i 3796 . . . . . . . . . 10  |-  ( (/)  e.  om  ->  om  =/=  (/) )
86, 7mp1i 12 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  om  =/=  (/) )
95, 8eqnetrd 2760 . . . . . . . 8  |-  ( f : om --> ~P A  ->  dom  f  =/=  (/) )
10 dm0rn0 5225 . . . . . . . . 9  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
1110necon3bii 2735 . . . . . . . 8  |-  ( dom  f  =/=  (/)  <->  ran  f  =/=  (/) )
129, 11sylib 196 . . . . . . 7  |-  ( f : om --> ~P A  ->  ran  f  =/=  (/) )
1312ad2antrl 727 . . . . . 6  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  ran  f  =/=  (/) )
14 ffn 5737 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  f  Fn  om )
1514ad2antrl 727 . . . . . . . 8  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  f  Fn  om )
16 sspss 3608 . . . . . . . . . . 11  |-  ( ( f `  suc  b
)  C_  ( f `  b )  <->  ( (
f `  suc  b ) 
C.  ( f `  b )  \/  (
f `  suc  b )  =  ( f `  b ) ) )
17 fvex 5882 . . . . . . . . . . . . . . 15  |-  ( f `
 b )  e. 
_V
18 fvex 5882 . . . . . . . . . . . . . . 15  |-  ( f `
 suc  b )  e.  _V
1917, 18brcnv 5191 . . . . . . . . . . . . . 14  |-  ( ( f `  b ) `' [ C.]  ( f `  suc  b )  <->  ( f `  suc  b ) [ C.]  ( f `  b
) )
2017brrpss 6578 . . . . . . . . . . . . . 14  |-  ( ( f `  suc  b
) [ C.]  ( f `  b )  <->  ( f `  suc  b )  C.  ( f `  b
) )
2119, 20bitri 249 . . . . . . . . . . . . 13  |-  ( ( f `  b ) `' [ C.]  ( f `  suc  b )  <->  ( f `  suc  b )  C.  ( f `  b
) )
22 eqcom 2476 . . . . . . . . . . . . 13  |-  ( ( f `  b )  =  ( f `  suc  b )  <->  ( f `  suc  b )  =  ( f `  b
) )
2321, 22orbi12i 521 . . . . . . . . . . . 12  |-  ( ( ( f `  b
) `' [ C.]  (
f `  suc  b )  \/  ( f `  b )  =  ( f `  suc  b
) )  <->  ( (
f `  suc  b ) 
C.  ( f `  b )  \/  (
f `  suc  b )  =  ( f `  b ) ) )
2423biimpri 206 . . . . . . . . . . 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 195 . . . . . . . . . 10  |-  ( ( f `  suc  b
)  C_  ( f `  b )  ->  (
( f `  b
) `' [ C.]  (
f `  suc  b )  \/  ( f `  b )  =  ( f `  suc  b
) ) )
2625ralimi 2860 . . . . . . . . 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 728 . . . . . . . 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 6579 . . . . . . . . . 10  |- [ C.]  Po  ran  f
29 cnvpo 5551 . . . . . . . . . 10  |-  ( [ C.]  Po  ran  f  <->  `' [ C.]  Po  ran  f )
3028, 29mpbi 208 . . . . . . . . 9  |-  `' [ C.]  Po  ran  f
3130a1i 11 . . . . . . . 8  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  `' [ C.]  Po  ran  f )
32 sornom 8669 . . . . . . . 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 1228 . . . . . . 7  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  `' [ C.]  Or  ran  f )
34 cnvso 5552 . . . . . . 7  |-  ( [ C.]  Or  ran  f  <->  `' [ C.]  Or  ran  f )
3533, 34sylibr 212 . . . . . 6  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  -> [ C.]  Or  ran  f
)
36 fin2i2 8710 . . . . . 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 1229 . . . . 5  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  |^| ran  f  e. 
ran  f )
3837expr 615 . . . 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 474 . . 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 2881 . 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 8721 . 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 232 1  |-  ( A  e. FinII  ->  A  e.  F
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2817    C_ wss 3481    C. wpss 3482   (/)c0 3790   ~Pcpw 4016   |^|cint 4288   class class class wbr 4453    Po wpo 4804    Or wor 4805   suc csuc 4886   `'ccnv 5004   dom cdm 5005   ran crn 5006    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   [ C.] crpss 6574   omcom 6695    ^m cmap 7432  FinIIcfin2 8671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-rpss 6575  df-om 6696  df-1st 6795  df-2nd 6796  df-map 7434  df-fin2 8678
This theorem is referenced by:  fin23  8781
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