MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem40 Structured version   Unicode version

Theorem fin23lem40 8512
Description: Lemma for fin23 8550. 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 7226 . . . 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 5558 . . . . . . 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 5556 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  dom  f  =  om )
6 peano1 6490 . . . . . . . . . 10  |-  (/)  e.  om
7 ne0i 3636 . . . . . . . . . 10  |-  ( (/)  e.  om  ->  om  =/=  (/) )
86, 7mp1i 12 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  om  =/=  (/) )
95, 8eqnetrd 2620 . . . . . . . 8  |-  ( f : om --> ~P A  ->  dom  f  =/=  (/) )
10 dm0rn0 5048 . . . . . . . . 9  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
1110necon3bii 2634 . . . . . . . 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 5552 . . . . . . . . 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 3448 . . . . . . . . . . 11  |-  ( ( f `  suc  b
)  C_  ( f `  b )  <->  ( (
f `  suc  b ) 
C.  ( f `  b )  \/  (
f `  suc  b )  =  ( f `  b ) ) )
17 fvex 5694 . . . . . . . . . . . . . . 15  |-  ( f `
 b )  e. 
_V
18 fvex 5694 . . . . . . . . . . . . . . 15  |-  ( f `
 suc  b )  e.  _V
1917, 18brcnv 5014 . . . . . . . . . . . . . 14  |-  ( ( f `  b ) `' [ C.]  ( f `  suc  b )  <->  ( f `  suc  b ) [ C.]  ( f `  b
) )
2017brrpss 6358 . . . . . . . . . . . . . 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 2439 . . . . . . . . . . . . 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 2785 . . . . . . . . 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 6359 . . . . . . . . . 10  |- [ C.]  Po  ran  f
29 cnvpo 5368 . . . . . . . . . 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 8438 . . . . . . . 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 1218 . . . . . . 7  |-  ( ( A  e. FinII  /\  ( f : om --> ~P A  /\  A. b  e.  om  (
f `  suc  b ) 
C_  ( f `  b ) ) )  ->  `' [ C.]  Or  ran  f )
34 cnvso 5369 . . . . . . 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 8479 . . . . . 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 1219 . . . . 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 2793 . 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 8490 . 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 1369    e. wcel 1756   {cab 2423    =/= wne 2600   A.wral 2709    C_ wss 3321    C. wpss 3322   (/)c0 3630   ~Pcpw 3853   |^|cint 4121   class class class wbr 4285    Po wpo 4631    Or wor 4632   suc csuc 4713   `'ccnv 4831   dom cdm 4832   ran crn 4833    Fn wfn 5406   -->wf 5407   ` cfv 5411  (class class class)co 6086   [ C.] crpss 6354   omcom 6471    ^m cmap 7206  FinIIcfin2 8440
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 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367
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 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-fv 5419  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-rpss 6355  df-om 6472  df-1st 6572  df-2nd 6573  df-map 7208  df-fin2 8447
This theorem is referenced by:  fin23  8550
  Copyright terms: Public domain W3C validator