Theorem fin23lem40 9173

Description: Lemma for fin23 9211. Fin^II sets satisfy the descending chain condition. (Contributed by Stefan O'Rear, 3-Nov-2014.)

Hypothesis

Ref	Expression
fin23lem40.f	|- F = { g | A. a e. ( ~P g ^m om ) ( A. x e. om ( a ` suc x ) C_ ( a ` x ) -> |^| ran a e. ran a ) }

Assertion

Ref	Expression
fin23lem40	|- ( A e. FinII -> A e. F )

Distinct variable groups:   g, a, x, A   F, a

Allowed substitution hints:   F( x, g)

Proof of Theorem fin23lem40

Dummy variables b f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 7879	. . . 4 |- ( f e. ( ~P A ^m om ) -> f : om --> ~P A )
2		simpl 473	. . . . . 6 |- ( ( A e. FinII /\ ( f : om --> ~P A /\ A. b e. om ( f ` suc b ) C_ ( f ` b ) ) ) -> A e. FinII )
3		frn 6053	. . . . . . 7 |- ( f : om --> ~P A -> ran f C_ ~P A )
4	3	ad2antrl 764	. . . . . 6 |- ( ( A e. FinII /\ ( f : om --> ~P A /\ A. b e. om ( f ` suc b ) C_ ( f ` b ) ) ) -> ran f C_ ~P A )
5		fdm 6051	. . . . . . . . 9 |- ( f : om --> ~P A -> dom f = om )
6		peano1 7085	. . . . . . . . . 10 |- (/) e. om
7		ne0i 3921	. . . . . . . . . 10 |- ( (/) e. om -> om =/= (/) )
8	6, 7	mp1i 13	. . . . . . . . 9 |- ( f : om --> ~P A -> om =/= (/) )
9	5, 8	eqnetrd 2861	. . . . . . . 8 |- ( f : om --> ~P A -> dom f =/= (/) )
10		dm0rn0 5342	. . . . . . . . 9 |- ( dom f = (/) <-> ran f = (/) )
11	10	necon3bii 2846	. . . . . . . 8 |- ( dom f =/= (/) <-> ran f =/= (/) )
12	9, 11	sylib 208	. . . . . . 7 |- ( f : om --> ~P A -> ran f =/= (/) )
13	12	ad2antrl 764	. . . . . 6 |- ( ( A e. FinII /\ ( f : om --> ~P A /\ A. b e. om ( f ` suc b ) C_ ( f ` b ) ) ) -> ran f =/= (/) )
14		ffn 6045	. . . . . . . . 9 |- ( f : om --> ~P A -> f Fn om )
15	14	ad2antrl 764	. . . . . . . 8 |- ( ( A e. FinII /\ ( f : om --> ~P A /\ A. b e. om ( f ` suc b ) C_ ( f ` b ) ) ) -> f Fn om )
16		sspss 3706	. . . . . . . . . . 11 |- ( ( f ` suc b ) C_ ( f ` b ) <-> ( ( f ` suc b ) C. ( f ` b ) \/ ( f ` suc b ) = ( f ` b ) ) )
17		fvex 6201	. . . . . . . . . . . . . 14 |- ( f ` b ) e. _V
18		fvex 6201	. . . . . . . . . . . . . 14 |- ( f ` suc b ) e. _V
19	17, 18	brcnv 5305	. . . . . . . . . . . . 13 |- ( ( f ` b ) `' [ C.] ( f ` suc b ) <-> ( f ` suc b ) [ C.] ( f ` b ) )
20	17	brrpss 6940	. . . . . . . . . . . . 13 |- ( ( f ` suc b ) [ C.] ( f ` b ) <-> ( f ` suc b ) C. ( f ` b ) )
21	19, 20	bitri 264	. . . . . . . . . . . 12 |- ( ( f ` b ) `' [ C.] ( f ` suc b ) <-> ( f ` suc b ) C. ( f ` b ) )
22		eqcom 2629	. . . . . . . . . . . 12 |- ( ( f ` b ) = ( f ` suc b ) <-> ( f ` suc b ) = ( f ` b ) )
23	21, 22	orbi12i 543	. . . . . . . . . . 11 |- ( ( ( f ` b ) `' [ C.] ( f ` suc b ) \/ ( f ` b ) = ( f ` suc b ) ) <-> ( ( f ` suc b ) C. ( f ` b ) \/ ( f ` suc b ) = ( f ` b ) ) )
24	16, 23	sylbb2 228	. . . . . . . . . 10 |- ( ( f ` suc b ) C_ ( f ` b ) -> ( ( f ` b ) `' [ C.] ( f ` suc b ) \/ ( f ` b ) = ( f ` suc b ) ) )
25	24	ralimi 2952	. . . . . . . . 9 |- ( A. b e. om ( f ` suc b ) C_ ( f ` b ) -> A. b e. om ( ( f ` b ) `' [ C.] ( f ` suc b ) \/ ( f ` b ) = ( f ` suc b ) ) )
26	25	ad2antll 765	. . . . . . . 8 |- ( ( A e. FinII /\ ( f : om --> ~P A /\ A. b e. om ( f ` suc b ) C_ ( f ` b ) ) ) -> A. b e. om ( ( f ` b ) `' [ C.] ( f ` suc b ) \/ ( f ` b ) = ( f ` suc b ) ) )
27		porpss 6941	. . . . . . . . . 10 |- [ C.] Po ran f
28		cnvpo 5673	. . . . . . . . . 10 |- ( [ C.] Po ran f <-> `' [ C.] Po ran f )
29	27, 28	mpbi 220	. . . . . . . . 9 |- `' [ C.] Po ran f
30	29	a1i 11	. . . . . . . 8 |- ( ( A e. FinII /\ ( f : om --> ~P A /\ A. b e. om ( f ` suc b ) C_ ( f ` b ) ) ) -> `' [ C.] Po ran f )
31		sornom 9099	. . . . . . . 8 |- ( ( f Fn om /\ A. b e. om ( ( f ` b ) `' [ C.] ( f ` suc b ) \/ ( f ` b ) = ( f ` suc b ) ) /\ `' [ C.] Po ran f ) -> `' [ C.] Or ran f )
32	15, 26, 30, 31	syl3anc 1326	. . . . . . 7 |- ( ( A e. FinII /\ ( f : om --> ~P A /\ A. b e. om ( f ` suc b ) C_ ( f ` b ) ) ) -> `' [ C.] Or ran f )
33		cnvso 5674	. . . . . . 7 |- ( [ C.] Or ran f <-> `' [ C.] Or ran f )
34	32, 33	sylibr 224	. . . . . 6 |- ( ( A e. FinII /\ ( f : om --> ~P A /\ A. b e. om ( f ` suc b ) C_ ( f ` b ) ) ) -> [ C.] Or ran f )
35		fin2i2 9140	. . . . . 6 |- ( ( ( A e. FinII /\ ran f C_ ~P A ) /\ ( ran f =/= (/) /\ [ C.] Or ran f ) ) -> |^| ran f e. ran f )
36	2, 4, 13, 34, 35	syl22anc 1327	. . . . 5 |- ( ( A e. FinII /\ ( f : om --> ~P A /\ A. b e. om ( f ` suc b ) C_ ( f ` b ) ) ) -> |^| ran f e. ran f )
37	36	expr 643	. . . 4 |- ( ( A e. FinII /\ f : om --> ~P A ) -> ( A. b e. om ( f ` suc b ) C_ ( f ` b ) -> |^| ran f e. ran f ) )
38	1, 37	sylan2 491	. . 3 |- ( ( A e. FinII /\ f e. ( ~P A ^m om ) ) -> ( A. b e. om ( f ` suc b ) C_ ( f ` b ) -> |^| ran f e. ran f ) )
39	38	ralrimiva 2966	. 2 |- ( A e. FinII -> A. f e. ( ~P A ^m om ) ( A. b e. om ( f ` suc b ) C_ ( f ` b ) -> |^| ran f e. ran f ) )
40		fin23lem40.f	. . 3 |- F = { g | A. a e. ( ~P g ^m om ) ( A. x e. om ( a ` suc x ) C_ ( a ` x ) -> |^| ran a e. ran a ) }
41	40	isfin3ds 9151	. 2 |- ( A e. FinII -> ( A e. F <-> A. f e. ( ~P A ^m om ) ( A. b e. om ( f ` suc b ) C_ ( f ` b ) -> |^| ran f e. ran f ) ) )
42	39, 41	mpbird 247	1 |- ( A e. FinII -> A e. F )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   C_ wss 3574   C. wpss 3575   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   class class class wbr 4653   Po wpo 5033   Or wor 5034   `'ccnv 5113   dom cdm 5114   ran crn 5115   suc csuc 5725   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   [ C.] crpss 6936   omcom 7065   ^m cmap 7857  Fin^IIcfin2 9101

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rpss 6937  df-om 7066  df-1st 7168  df-2nd 7169  df-map 7859  df-fin2 9108

This theorem is referenced by:  fin23  9211