Theorem isfin3ds 8705

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 4943	. . . . . . . . 9 |- ( b = x -> suc b = suc x )
2	1	fveq2d 5868	. . . . . . . 8 |- ( b = x -> ( a ` suc b ) = ( a ` suc x ) )
3		fveq2 5864	. . . . . . . 8 |- ( b = x -> ( a ` b ) = ( a ` x ) )
4	2, 3	sseq12d 3533	. . . . . . 7 |- ( b = x -> ( ( a ` suc b ) C_ ( a ` b ) <-> ( a ` suc x ) C_ ( a ` x ) ) )
5	4	cbvralv 3088	. . . . . 6 |- ( A. b e. om ( a ` suc b ) C_ ( a ` b ) <-> A. x e. om ( a ` suc x ) C_ ( a ` x ) )
6		fveq1 5863	. . . . . . . 8 |- ( a = f -> ( a ` suc x ) = ( f ` suc x ) )
7		fveq1 5863	. . . . . . . 8 |- ( a = f -> ( a ` x ) = ( f ` x ) )
8	6, 7	sseq12d 3533	. . . . . . 7 |- ( a = f -> ( ( a ` suc x ) C_ ( a ` x ) <-> ( f ` suc x ) C_ ( f ` x ) ) )
9	8	ralbidv 2903	. . . . . 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 5226	. . . . . . 7 |- ( a = f -> ran a = ran f )
12	11	inteqd 4287	. . . . . 6 |- ( a = f -> |^| ran a = |^| ran f )
13	12, 11	eleq12d 2549	. . . . 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 3088	. . 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 4013	. . . . 5 |- ( g = A -> ~P g = ~P A )
17	16	oveq1d 6297	. . . 4 |- ( g = A -> ( ~P g ^m om ) = ( ~P A ^m om ) )
18	17	raleqdv 3064	. . 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 3252	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 1379   e. wcel 1767   {cab 2452   A.wral 2814   C_ wss 3476   ~Pcpw 4010   |^|cint 4282   suc csuc 4880   ran crn 5000   ` cfv 5586  (class class class)co 6282   omcom 6678   ^m cmap 7417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-suc 4884  df-cnv 5007  df-dm 5009  df-rn 5010  df-iota 5549  df-fv 5594  df-ov 6285

This theorem is referenced by:  ssfin3ds  8706  fin23lem17  8714  fin23lem39  8726  fin23lem40  8727  isf32lem12  8740  isfin3-3  8744