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Theorem ssfin3ds 8504
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 4481 . . . . 5  |-  ( A  e.  F  ->  ~P A  e.  _V )
21adantr 465 . . . 4  |-  ( ( A  e.  F  /\  B  C_  A )  ->  ~P A  e.  _V )
3 simpr 461 . . . . 5  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  C_  A )
4 sspwb 4546 . . . . 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 7260 . . . 4  |-  ( ( ~P A  e.  _V  /\ 
~P B  C_  ~P A )  ->  ( ~P B  ^m  om )  C_  ( ~P A  ^m  om ) )
72, 5, 6syl2anc 661 . . 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 8503 . . . . 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 465 . . 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 3421 . . 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 4443 . . . 4  |-  ( ( B  C_  A  /\  A  e.  F )  ->  B  e.  _V )
1514ancoms 453 . . 3  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  e.  _V )
168isfin3ds 8503 . . 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 1369    e. wcel 1756   {cab 2429   A.wral 2720   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865   |^|cint 4133   suc csuc 4726   ran crn 4846   ` cfv 5423  (class class class)co 6096   omcom 6481    ^m cmap 7219
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-map 7221
This theorem is referenced by:  fin23lem31  8517
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