Theorem isfin3ds 8700

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 4932	. . . . . . . . 9 |- ( b = x -> suc b = suc x )
2	1	fveq2d 5852	. . . . . . . 8 |- ( b = x -> ( a ` suc b ) = ( a ` suc x ) )
3		fveq2 5848	. . . . . . . 8 |- ( b = x -> ( a ` b ) = ( a ` x ) )
4	2, 3	sseq12d 3518	. . . . . . 7 |- ( b = x -> ( ( a ` suc b ) C_ ( a ` b ) <-> ( a ` suc x ) C_ ( a ` x ) ) )
5	4	cbvralv 3081	. . . . . 6 |- ( A. b e. om ( a ` suc b ) C_ ( a ` b ) <-> A. x e. om ( a ` suc x ) C_ ( a ` x ) )
6		fveq1 5847	. . . . . . . 8 |- ( a = f -> ( a ` suc x ) = ( f ` suc x ) )
7		fveq1 5847	. . . . . . . 8 |- ( a = f -> ( a ` x ) = ( f ` x ) )
8	6, 7	sseq12d 3518	. . . . . . 7 |- ( a = f -> ( ( a ` suc x ) C_ ( a ` x ) <-> ( f ` suc x ) C_ ( f ` x ) ) )
9	8	ralbidv 2893	. . . . . 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 5217	. . . . . . 7 |- ( a = f -> ran a = ran f )
12	11	inteqd 4276	. . . . . 6 |- ( a = f -> |^| ran a = |^| ran f )
13	12, 11	eleq12d 2536	. . . . 5 |- ( a = f -> ( |^| ran a e. ran a <-> |^| ran f e. ran f ) )
14	10, 13	imbi12d 318	. . . 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 3081	. . 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 4002	. . . . 5 |- ( g = A -> ~P g = ~P A )
17	16	oveq1d 6285	. . . 4 |- ( g = A -> ( ~P g ^m om ) = ( ~P A ^m om ) )
18	17	raleqdv 3057	. . 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 3245	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 1398   e. wcel 1823   {cab 2439   A.wral 2804   C_ wss 3461   ~Pcpw 3999   |^|cint 4271   suc csuc 4869   ran crn 4989   ` cfv 5570  (class class class)co 6270   omcom 6673   ^m cmap 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-suc 4873  df-cnv 4996  df-dm 4998  df-rn 4999  df-iota 5534  df-fv 5578  df-ov 6273

This theorem is referenced by:  ssfin3ds  8701  fin23lem17  8709  fin23lem39  8721  fin23lem40  8722  isf32lem12  8735  isfin3-3  8739