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Theorem ssfin3ds 7956
Description: A subset of a III-finite set is III-finite. (Contributed by Stefan O'Rear, 4-Nov-2014.)
Hypothesis
Ref Expression
isfin3ds.f  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. b  e.  om  ( a `  suc  b )  C_  (
a `  b )  ->  |^| ran  a  e. 
ran  a ) }
Assertion
Ref Expression
ssfin3ds  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  e.  F )
Distinct variable groups:    a, b,
g, A    B, a,
b, g
Allowed substitution hints:    F( g, a, b)

Proof of Theorem ssfin3ds
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4194 . . . . 5  |-  ( A  e.  F  ->  ~P A  e.  _V )
21adantr 451 . . . 4  |-  ( ( A  e.  F  /\  B  C_  A )  ->  ~P A  e.  _V )
3 simpr 447 . . . . 5  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  C_  A )
4 sspwb 4223 . . . . 5  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
53, 4sylib 188 . . . 4  |-  ( ( A  e.  F  /\  B  C_  A )  ->  ~P B  C_  ~P A
)
6 mapss 6810 . . . 4  |-  ( ( ~P A  e.  _V  /\ 
~P B  C_  ~P A )  ->  ( ~P B  ^m  om )  C_  ( ~P A  ^m  om ) )
72, 5, 6syl2anc 642 . . 3  |-  ( ( A  e.  F  /\  B  C_  A )  -> 
( ~P B  ^m  om )  C_  ( ~P A  ^m  om ) )
8 isfin3ds.f . . . . . 6  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. b  e.  om  ( a `  suc  b )  C_  (
a `  b )  ->  |^| ran  a  e. 
ran  a ) }
98isfin3ds 7955 . . . . 5  |-  ( A  e.  F  ->  ( A  e.  F  <->  A. f  e.  ( ~P A  ^m  om ) ( A. x  e.  om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) ) )
109ibi 232 . . . 4  |-  ( A  e.  F  ->  A. f  e.  ( ~P A  ^m  om ) ( A. x  e.  om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) )
1110adantr 451 . . 3  |-  ( ( A  e.  F  /\  B  C_  A )  ->  A. f  e.  ( ~P A  ^m  om )
( A. x  e. 
om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) )
12 ssralv 3237 . . 3  |-  ( ( ~P B  ^m  om )  C_  ( ~P A  ^m  om )  ->  ( A. f  e.  ( ~P A  ^m  om )
( A. x  e. 
om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f )  ->  A. f  e.  ( ~P B  ^m  om )
( A. x  e. 
om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) ) )
137, 11, 12sylc 56 . 2  |-  ( ( A  e.  F  /\  B  C_  A )  ->  A. f  e.  ( ~P B  ^m  om )
( A. x  e. 
om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) )
14 ssexg 4160 . . . 4  |-  ( ( B  C_  A  /\  A  e.  F )  ->  B  e.  _V )
1514ancoms 439 . . 3  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  e.  _V )
168isfin3ds 7955 . . 3  |-  ( B  e.  _V  ->  ( B  e.  F  <->  A. f  e.  ( ~P B  ^m  om ) ( A. x  e.  om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) ) )
1715, 16syl 15 . 2  |-  ( ( A  e.  F  /\  B  C_  A )  -> 
( B  e.  F  <->  A. f  e.  ( ~P B  ^m  om )
( A. x  e. 
om  ( f `  suc  x )  C_  (
f `  x )  ->  |^| ran  f  e. 
ran  f ) ) )
1813, 17mpbird 223 1  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   |^|cint 3862   suc csuc 4394   omcom 4656   ran crn 4690   ` cfv 5255  (class class class)co 5858    ^m cmap 6772
This theorem is referenced by:  fin23lem31  7969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774
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