Theorem isfin3ds 8498

Description: Property of a III-finite set (descending sequence version). (Contributed by Mario Carneiro, 16-May-2015.)

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
isfin3ds	|- ( A e. V -> ( 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 ) ) )

Distinct variable group:   a, b, f, g, x, A

Allowed substitution hints:   F( x, f, g, a, b)   V( x, f, g, a, b)

Proof of Theorem isfin3ds

Step	Hyp	Ref	Expression
1		suceq 4784	. . . . . . . . 9 |- ( b = x -> suc b = suc x )
2	1	fveq2d 5695	. . . . . . . 8 |- ( b = x -> ( a ` suc b ) = ( a ` suc x ) )
3		fveq2 5691	. . . . . . . 8 |- ( b = x -> ( a ` b ) = ( a ` x ) )
4	2, 3	sseq12d 3385	. . . . . . 7 |- ( b = x -> ( ( a ` suc b ) C_ ( a ` b ) <-> ( a ` suc x ) C_ ( a ` x ) ) )
5	4	cbvralv 2947	. . . . . 6 |- ( A. b e. om ( a ` suc b ) C_ ( a ` b ) <-> A. x e. om ( a ` suc x ) C_ ( a ` x ) )
6		fveq1 5690	. . . . . . . 8 |- ( a = f -> ( a ` suc x ) = ( f ` suc x ) )
7		fveq1 5690	. . . . . . . 8 |- ( a = f -> ( a ` x ) = ( f ` x ) )
8	6, 7	sseq12d 3385	. . . . . . 7 |- ( a = f -> ( ( a ` suc x ) C_ ( a ` x ) <-> ( f ` suc x ) C_ ( f ` x ) ) )
9	8	ralbidv 2735	. . . . . 6 |- ( a = f -> ( A. x e. om ( a ` suc x ) C_ ( a ` x ) <-> A. x e. om ( f ` suc x ) C_ ( f ` x ) ) )
10	5, 9	syl5bb 257	. . . . 5 |- ( a = f -> ( A. b e. om ( a ` suc b ) C_ ( a ` b ) <-> A. x e. om ( f ` suc x ) C_ ( f ` x ) ) )
11		rneq 5065	. . . . . . 7 |- ( a = f -> ran a = ran f )
12	11	inteqd 4133	. . . . . 6 |- ( a = f -> |^| ran a = |^| ran f )
13	12, 11	eleq12d 2511	. . . . 5 |- ( a = f -> ( |^| ran a e. ran a <-> |^| ran f e. ran f ) )
14	10, 13	imbi12d 320	. . . 4 |- ( a = f -> ( ( A. b e. om ( a ` suc b ) C_ ( a ` b ) -> |^| ran a e. ran a ) <-> ( A. x e. om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) ) )
15	14	cbvralv 2947	. . 3 |- ( A. a e. ( ~P g ^m om ) ( A. b e. om ( a ` suc b ) C_ ( a ` b ) -> |^| ran a e. ran a ) <-> A. f e. ( ~P g ^m om ) ( A. x e. om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) )
16		pweq 3863	. . . . 5 |- ( g = A -> ~P g = ~P A )
17	16	oveq1d 6106	. . . 4 |- ( g = A -> ( ~P g ^m om ) = ( ~P A ^m om ) )
18	17	raleqdv 2923	. . 3 |- ( g = A -> ( A. f e. ( ~P g ^m om ) ( A. x e. om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) <-> A. f e. ( ~P A ^m om ) ( A. x e. om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) ) )
19	15, 18	syl5bb 257	. 2 |- ( g = A -> ( A. a e. ( ~P g ^m om ) ( A. b e. om ( a ` suc b ) C_ ( a ` b ) -> |^| ran a e. ran a ) <-> A. f e. ( ~P A ^m om ) ( A. x e. om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) ) )
20		isfin3ds.f	. 2 |- F = { g | A. a e. ( ~P g ^m om ) ( A. b e. om ( a ` suc b ) C_ ( a ` b ) -> |^| ran a e. ran a ) }
21	19, 20	elab2g 3108	1 |- ( A e. V -> ( 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 ) ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1369   e. wcel 1756   {cab 2429   A.wral 2715   C_ wss 3328   ~Pcpw 3860   |^|cint 4128   suc csuc 4721   ran crn 4841   ` cfv 5418  (class class class)co 6091   omcom 6476   ^m cmap 7214

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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

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-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-suc 4725  df-cnv 4848  df-dm 4850  df-rn 4851  df-iota 5381  df-fv 5426  df-ov 6094

This theorem is referenced by:  ssfin3ds  8499  fin23lem17  8507  fin23lem39  8519  fin23lem40  8520  isf32lem12  8533  isfin3-3  8537