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Theorem fin23lem40 8231
Description: Lemma for fin23 8269. 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 7038 . . . 4  |-  ( f  e.  ( ~P A  ^m  om )  ->  f : om --> ~P A )
2 simpl 444 . . . . . 6  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  A  e. FinII )
3 frn 5597 . . . . . . 7  |-  ( f : om --> ~P A  ->  ran  f  C_  ~P A )
43ad2antrl 709 . . . . . 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 5595 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  dom  f  =  om )
6 peano1 4864 . . . . . . . . . 10  |-  (/)  e.  om
7 ne0i 3634 . . . . . . . . . 10  |-  ( (/)  e.  om  ->  om  =/=  (/) )
86, 7mp1i 12 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  om  =/=  (/) )
95, 8eqnetrd 2619 . . . . . . . 8  |-  ( f : om --> ~P A  ->  dom  f  =/=  (/) )
10 dm0rn0 5086 . . . . . . . . 9  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
1110necon3bii 2633 . . . . . . . 8  |-  ( dom  f  =/=  (/)  <->  ran  f  =/=  (/) )
129, 11sylib 189 . . . . . . 7  |-  ( f : om --> ~P A  ->  ran  f  =/=  (/) )
1312ad2antrl 709 . . . . . 6  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  ran  f  =/=  (/) )
14 ffn 5591 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  f  Fn  om )
1514ad2antrl 709 . . . . . . . 8  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  f  Fn  om )
16 sspss 3446 . . . . . . . . . . 11  |-  ( ( f `  suc  b
)  C_  ( f `  b )  <->  ( (
f `  suc  b ) 
C.  ( f `  b )  \/  (
f `  suc  b )  =  ( f `  b ) ) )
17 fvex 5742 . . . . . . . . . . . . . . 15  |-  ( f `
 b )  e. 
_V
18 fvex 5742 . . . . . . . . . . . . . . 15  |-  ( f `
 suc  b )  e.  _V
1917, 18brcnv 5055 . . . . . . . . . . . . . 14  |-  ( ( f `  b ) `' [ C.]  ( f `  suc  b )  <->  ( f `  suc  b ) [ C.]  ( f `  b
) )
2017brrpss 6525 . . . . . . . . . . . . . 14  |-  ( ( f `  suc  b
) [ C.]  ( f `  b )  <->  ( f `  suc  b )  C.  ( f `  b
) )
2119, 20bitri 241 . . . . . . . . . . . . 13  |-  ( ( f `  b ) `' [ C.]  ( f `  suc  b )  <->  ( f `  suc  b )  C.  ( f `  b
) )
22 eqcom 2438 . . . . . . . . . . . . 13  |-  ( ( f `  b )  =  ( f `  suc  b )  <->  ( f `  suc  b )  =  ( f `  b
) )
2321, 22orbi12i 508 . . . . . . . . . . . 12  |-  ( ( ( f `  b
) `' [ C.]  (
f `  suc  b )  \/  ( f `  b )  =  ( f `  suc  b
) )  <->  ( (
f `  suc  b ) 
C.  ( f `  b )  \/  (
f `  suc  b )  =  ( f `  b ) ) )
2423biimpri 198 . . . . . . . . . . 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 188 . . . . . . . . . 10  |-  ( ( f `  suc  b
)  C_  ( f `  b )  ->  (
( f `  b
) `' [ C.]  (
f `  suc  b )  \/  ( f `  b )  =  ( f `  suc  b
) ) )
2625ralimi 2781 . . . . . . . . 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 710 . . . . . . . 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 6526 . . . . . . . . . 10  |- [ C.]  Po  ran  f
29 cnvpo 5410 . . . . . . . . . 10  |-  ( [ C.]  Po  ran  f  <->  `' [ C.]  Po  ran  f )
3028, 29mpbi 200 . . . . . . . . 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 8157 . . . . . . . 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 1184 . . . . . . 7  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  `' [ C.]  Or  ran  f )
34 cnvso 5411 . . . . . . 7  |-  ( [ C.]  Or  ran  f  <->  `' [ C.]  Or  ran  f )
3533, 34sylibr 204 . . . . . 6  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  -> [ C.]  Or  ran  f )
36 fin2i2 8198 . . . . . 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 1185 . . . . 5  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  |^| ran  f  e. 
ran  f )
3837expr 599 . . . 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 461 . . 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 2789 . 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 8209 . 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 224 1  |-  ( A  e. FinII  ->  A  e.  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705    C_ wss 3320    C. wpss 3321   (/)c0 3628   ~Pcpw 3799   |^|cint 4050   class class class wbr 4212    Po wpo 4501    Or wor 4502   suc csuc 4583   omcom 4845   `'ccnv 4877   dom cdm 4878   ran crn 4879    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   [ C.] crpss 6521    ^m cmap 7018  FinIIcfin2 8159
This theorem is referenced by:  fin23  8269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-rpss 6522  df-map 7020  df-fin2 8166
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