Theorem isfin3ds 9151

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 5790	. . . . . . . . 9 |- ( b = x -> suc b = suc x )
2	1	fveq2d 6195	. . . . . . . 8 |- ( b = x -> ( a ` suc b ) = ( a ` suc x ) )
3		fveq2 6191	. . . . . . . 8 |- ( b = x -> ( a ` b ) = ( a ` x ) )
4	2, 3	sseq12d 3634	. . . . . . 7 |- ( b = x -> ( ( a ` suc b ) C_ ( a ` b ) <-> ( a ` suc x ) C_ ( a ` x ) ) )
5	4	cbvralv 3171	. . . . . 6 |- ( A. b e. om ( a ` suc b ) C_ ( a ` b ) <-> A. x e. om ( a ` suc x ) C_ ( a ` x ) )
6		fveq1 6190	. . . . . . . 8 |- ( a = f -> ( a ` suc x ) = ( f ` suc x ) )
7		fveq1 6190	. . . . . . . 8 |- ( a = f -> ( a ` x ) = ( f ` x ) )
8	6, 7	sseq12d 3634	. . . . . . 7 |- ( a = f -> ( ( a ` suc x ) C_ ( a ` x ) <-> ( f ` suc x ) C_ ( f ` x ) ) )
9	8	ralbidv 2986	. . . . . 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 272	. . . . 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 5351	. . . . . . 7 |- ( a = f -> ran a = ran f )
12	11	inteqd 4480	. . . . . 6 |- ( a = f -> |^| ran a = |^| ran f )
13	12, 11	eleq12d 2695	. . . . 5 |- ( a = f -> ( |^| ran a e. ran a <-> |^| ran f e. ran f ) )
14	10, 13	imbi12d 334	. . . 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 3171	. . 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 4161	. . . . 5 |- ( g = A -> ~P g = ~P A )
17	16	oveq1d 6665	. . . 4 |- ( g = A -> ( ~P g ^m om ) = ( ~P A ^m om ) )
18	17	raleqdv 3144	. . 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 272	. 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 3353	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 196   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-suc 5729  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  ssfin3ds  9152  fin23lem17  9160  fin23lem39  9172  fin23lem40  9173  isf32lem12  9186  isfin3-3  9190