MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssfin3ds Structured version   Unicode version

Theorem ssfin3ds 8202
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 4375 . . . . 5  |-  ( A  e.  F  ->  ~P A  e.  _V )
21adantr 452 . . . 4  |-  ( ( A  e.  F  /\  B  C_  A )  ->  ~P A  e.  _V )
3 simpr 448 . . . . 5  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  C_  A )
4 sspwb 4405 . . . . 5  |-  ( B 
C_  A  <->  ~P B  C_ 
~P A )
53, 4sylib 189 . . . 4  |-  ( ( A  e.  F  /\  B  C_  A )  ->  ~P B  C_  ~P A
)
6 mapss 7048 . . . 4  |-  ( ( ~P A  e.  _V  /\ 
~P B  C_  ~P A )  ->  ( ~P B  ^m  om )  C_  ( ~P A  ^m  om ) )
72, 5, 6syl2anc 643 . . 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 8201 . . . . 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 233 . . . 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 452 . . 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 3399 . . 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 58 . 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 4341 . . . 4  |-  ( ( B  C_  A  /\  A  e.  F )  ->  B  e.  _V )
1514ancoms 440 . . 3  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  e.  _V )
168isfin3ds 8201 . . 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 224 1  |-  ( ( A  e.  F  /\  B  C_  A )  ->  B  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   |^|cint 4042   suc csuc 4575   omcom 4837   ran crn 4871   ` cfv 5446  (class class class)co 6073    ^m cmap 7010
This theorem is referenced by:  fin23lem31  8215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012
  Copyright terms: Public domain W3C validator