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Theorem fin23lem40 8541
Description: Lemma for fin23 8579. 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 7255 . . . 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 5586 . . . . . . 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 5584 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  dom  f  =  om )
6 peano1 6516 . . . . . . . . . 10  |-  (/)  e.  om
7 ne0i 3664 . . . . . . . . . 10  |-  ( (/)  e.  om  ->  om  =/=  (/) )
86, 7mp1i 12 . . . . . . . . 9  |-  ( f : om --> ~P A  ->  om  =/=  (/) )
95, 8eqnetrd 2654 . . . . . . . 8  |-  ( f : om --> ~P A  ->  dom  f  =/=  (/) )
10 dm0rn0 5077 . . . . . . . . 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 5580 . . . . . . . . 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 3476 . . . . . . . . . . 11  |-  ( ( f `  suc  b
)  C_  ( f `  b )  <->  ( (
f `  suc  b ) 
C.  ( f `  b )  \/  (
f `  suc  b )  =  ( f `  b ) ) )
17 fvex 5722 . . . . . . . . . . . . . . 15  |-  ( f `
 b )  e. 
_V
18 fvex 5722 . . . . . . . . . . . . . . 15  |-  ( f `
 suc  b )  e.  _V
1917, 18brcnv 5043 . . . . . . . . . . . . . 14  |-  ( ( f `  b ) `' [ C.]  ( f `  suc  b )  <->  ( f `  suc  b ) [ C.]  ( f `  b
) )
2017brrpss 6384 . . . . . . . . . . . . . 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 2445 . . . . . . . . . . . . 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 2812 . . . . . . . . 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 6385 . . . . . . . . . 10  |- [ C.]  Po  ran  f
29 cnvpo 5396 . . . . . . . . . 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 8467 . . . . . . . 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 5397 . . . . . . 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 8508 . . . . . 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 2820 . 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 8519 . 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 2429    =/= wne 2620   A.wral 2736    C_ wss 3349    C. wpss 3350   (/)c0 3658   ~Pcpw 3881   |^|cint 4149   class class class wbr 4313    Po wpo 4660    Or wor 4661   suc csuc 4742   `'ccnv 4860   dom cdm 4861   ran crn 4862    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   [ C.] crpss 6380   omcom 6497    ^m cmap 7235  FinIIcfin2 8469
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 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-rpss 6381  df-om 6498  df-1st 6598  df-2nd 6599  df-map 7237  df-fin2 8476
This theorem is referenced by:  fin23  8579
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