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Theorem ssfin3ds 8701
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 4621 . . . . 5  |-  ( A  e.  F  ->  ~P A  e.  _V )
21adantr 463 . . . 4  |-  ( ( A  e.  F  /\  B  C_  A )  ->  ~P A  e.  _V )
3 simpr 459 . . . . 5  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  C_  A )
4 sspwb 4686 . . . . 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 7454 . . . 4  |-  ( ( ~P A  e.  _V  /\ 
~P B  C_  ~P A )  ->  ( ~P B  ^m  om )  C_  ( ~P A  ^m  om ) )
72, 5, 6syl2anc 659 . . 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 8700 . . . . 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 463 . . 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 3550 . . 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 4583 . . . 4  |-  ( ( B  C_  A  /\  A  e.  F )  ->  B  e.  _V )
1514ancoms 451 . . 3  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  e.  _V )
168isfin3ds 8700 . . 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 367    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   |^|cint 4271   suc csuc 4869   ran crn 4989   ` cfv 5570  (class class class)co 6270   omcom 6673    ^m cmap 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414
This theorem is referenced by:  fin23lem31  8714
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