Theorem ssfin3ds 9152

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.

Step	Hyp	Ref	Expression
1		pwexg 4850	. . . . 5 |- ( A e. F -> ~P A e. _V )
2	1	adantr 481	. . . 4 |- ( ( A e. F /\ B C_ A ) -> ~P A e. _V )
3		simpr 477	. . . . 5 |- ( ( A e. F /\ B C_ A ) -> B C_ A )
4		sspwb 4917	. . . . 5 |- ( B C_ A <-> ~P B C_ ~P A )
5	3, 4	sylib 208	. . . 4 |- ( ( A e. F /\ B C_ A ) -> ~P B C_ ~P A )
6		mapss 7900	. . . 4 |- ( ( ~P A e. _V /\ ~P B C_ ~P A ) -> ( ~P B ^m om ) C_ ( ~P A ^m om ) )
7	2, 5, 6	syl2anc 693	. . 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 ) }
9	8	isfin3ds 9151	. . . . 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 ) ) )
10	9	ibi 256	. . . 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 ) )
11	10	adantr 481	. . 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 3666	. . 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 ) ) )
13	7, 11, 12	sylc 65	. 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 4804	. . . 4 |- ( ( B C_ A /\ A e. F ) -> B e. _V )
15	14	ancoms 469	. . 3 |- ( ( A e. F /\ B C_ A ) -> B e. _V )
16	8	isfin3ds 9151	. . 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 ) ) )
17	15, 16	syl 17	. 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 ) ) )
18	13, 17	mpbird 247	1 |- ( ( A e. F /\ B C_ A ) -> B e. F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |^|cint 4475   ran crn 5115   suc csuc 5725   ` cfv 5888  (class class class)co 6650   omcom 7065   ^m cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  fin23lem31  9165