Theorem isfin3ds 8777

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 5495	. . . . . . . . 9 |- ( b = x -> suc b = suc x )
2	1	fveq2d 5883	. . . . . . . 8 |- ( b = x -> ( a ` suc b ) = ( a ` suc x ) )
3		fveq2 5879	. . . . . . . 8 |- ( b = x -> ( a ` b ) = ( a ` x ) )
4	2, 3	sseq12d 3447	. . . . . . 7 |- ( b = x -> ( ( a ` suc b ) C_ ( a ` b ) <-> ( a ` suc x ) C_ ( a ` x ) ) )
5	4	cbvralv 3005	. . . . . 6 |- ( A. b e. om ( a ` suc b ) C_ ( a ` b ) <-> A. x e. om ( a ` suc x ) C_ ( a ` x ) )
6		fveq1 5878	. . . . . . . 8 |- ( a = f -> ( a ` suc x ) = ( f ` suc x ) )
7		fveq1 5878	. . . . . . . 8 |- ( a = f -> ( a ` x ) = ( f ` x ) )
8	6, 7	sseq12d 3447	. . . . . . 7 |- ( a = f -> ( ( a ` suc x ) C_ ( a ` x ) <-> ( f ` suc x ) C_ ( f ` x ) ) )
9	8	ralbidv 2829	. . . . . 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 265	. . . . 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 5066	. . . . . . 7 |- ( a = f -> ran a = ran f )
12	11	inteqd 4231	. . . . . 6 |- ( a = f -> |^| ran a = |^| ran f )
13	12, 11	eleq12d 2543	. . . . 5 |- ( a = f -> ( |^| ran a e. ran a <-> |^| ran f e. ran f ) )
14	10, 13	imbi12d 327	. . . 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 3005	. . 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 3945	. . . . 5 |- ( g = A -> ~P g = ~P A )
17	16	oveq1d 6323	. . . 4 |- ( g = A -> ( ~P g ^m om ) = ( ~P A ^m om ) )
18	17	raleqdv 2979	. . 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 265	. 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 3175	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 189   = wceq 1452   e. wcel 1904   {cab 2457   A.wral 2756   C_ wss 3390   ~Pcpw 3942   |^|cint 4226   ran crn 4840   suc csuc 5432   ` cfv 5589  (class class class)co 6308   omcom 6711   ^m cmap 7490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-cnv 4847  df-dm 4849  df-rn 4850  df-suc 5436  df-iota 5553  df-fv 5597  df-ov 6311

This theorem is referenced by:  ssfin3ds  8778  fin23lem17  8786  fin23lem39  8798  fin23lem40  8799  isf32lem12  8812  isfin3-3  8816