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Theorem ssfin3ds 8495
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 4473 . . . . 5  |-  ( A  e.  F  ->  ~P A  e.  _V )
21adantr 462 . . . 4  |-  ( ( A  e.  F  /\  B  C_  A )  ->  ~P A  e.  _V )
3 simpr 458 . . . . 5  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  C_  A )
4 sspwb 4538 . . . . 5  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
53, 4sylib 196 . . . 4  |-  ( ( A  e.  F  /\  B  C_  A )  ->  ~P B  C_  ~P A
)
6 mapss 7251 . . . 4  |-  ( ( ~P A  e.  _V  /\ 
~P B  C_  ~P A )  ->  ( ~P B  ^m  om )  C_  ( ~P A  ^m  om ) )
72, 5, 6syl2anc 656 . . 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 8494 . . . . 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 241 . . . 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 462 . . 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 3413 . . 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 60 . 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 4435 . . . 4  |-  ( ( B  C_  A  /\  A  e.  F )  ->  B  e.  _V )
1514ancoms 450 . . 3  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  e.  _V )
168isfin3ds 8494 . . 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 16 . 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 232 1  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  e.  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   {cab 2427   A.wral 2713   _Vcvv 2970    C_ wss 3325   ~Pcpw 3857   |^|cint 4125   suc csuc 4717   ran crn 4837   ` cfv 5415  (class class class)co 6090   omcom 6475    ^m cmap 7210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-map 7212
This theorem is referenced by:  fin23lem31  8508
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