Theorem isf32lem6 9180

Description: Lemma for isfin3-2 9189. Each K value is nonempty. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Hypotheses

Ref	Expression
isf32lem.a	|- ( ph -> F : om --> ~P G )
isf32lem.b	|- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )
isf32lem.c	|- ( ph -> -. |^| ran F e. ran F )
isf32lem.d	|- S = { y e. om | ( F ` suc y ) C. ( F ` y ) }
isf32lem.e	|- J = ( u e. om |-> ( iota_ v e. S ( v i^i S ) ~~ u ) )
isf32lem.f	|- K = ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) o. J )

Assertion

Ref	Expression
isf32lem6	|- ( ( ph /\ A e. om ) -> ( K ` A ) =/= (/) )

Distinct variable groups:   x, w   v, u, w, x, y, ph   w, A, x, y   w, F, x, y   u, S, v, w, x, y   w, J, x, y   x, K, y

Allowed substitution hints:   A( v, u)   F( v, u)   G( x, y, w, v, u)   J( v, u)   K( w, v, u)

Proof of Theorem isf32lem6

Step	Hyp	Ref	Expression
1		isf32lem.f	. . . 4 |- K = ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) o. J )
2	1	fveq1i 6192	. . 3 |- ( K ` A ) = ( ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) o. J ) ` A )
3		isf32lem.d	. . . . . . . 8 |- S = { y e. om | ( F ` suc y ) C. ( F ` y ) }
4		ssrab2 3687	. . . . . . . 8 |- { y e. om | ( F ` suc y ) C. ( F ` y ) } C_ om
5	3, 4	eqsstri 3635	. . . . . . 7 |- S C_ om
6		isf32lem.a	. . . . . . . 8 |- ( ph -> F : om --> ~P G )
7		isf32lem.b	. . . . . . . 8 |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )
8		isf32lem.c	. . . . . . . 8 |- ( ph -> -. |^| ran F e. ran F )
9	6, 7, 8, 3	isf32lem5 9179	. . . . . . 7 |- ( ph -> -. S e. Fin )
10		isf32lem.e	. . . . . . . 8 |- J = ( u e. om |-> ( iota_ v e. S ( v i^i S ) ~~ u ) )
11	10	fin23lem22 9149	. . . . . . 7 |- ( ( S C_ om /\ -. S e. Fin ) -> J : om -1-1-onto-> S )
12	5, 9, 11	sylancr 695	. . . . . 6 |- ( ph -> J : om -1-1-onto-> S )
13		f1of 6137	. . . . . 6 |- ( J : om -1-1-onto-> S -> J : om --> S )
14	12, 13	syl 17	. . . . 5 |- ( ph -> J : om --> S )
15		fvco3 6275	. . . . 5 |- ( ( J : om --> S /\ A e. om ) -> ( ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) o. J ) ` A ) = ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) ` ( J ` A ) ) )
16	14, 15	sylan 488	. . . 4 |- ( ( ph /\ A e. om ) -> ( ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) o. J ) ` A ) = ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) ` ( J ` A ) ) )
17	9	adantr 481	. . . . . . . 8 |- ( ( ph /\ A e. om ) -> -. S e. Fin )
18	5, 17, 11	sylancr 695	. . . . . . 7 |- ( ( ph /\ A e. om ) -> J : om -1-1-onto-> S )
19	18, 13	syl 17	. . . . . 6 |- ( ( ph /\ A e. om ) -> J : om --> S )
20		ffvelrn 6357	. . . . . 6 |- ( ( J : om --> S /\ A e. om ) -> ( J ` A ) e. S )
21	19, 20	sylancom 701	. . . . 5 |- ( ( ph /\ A e. om ) -> ( J ` A ) e. S )
22		fveq2 6191	. . . . . . 7 |- ( w = ( J ` A ) -> ( F ` w ) = ( F ` ( J ` A ) ) )
23		suceq 5790	. . . . . . . 8 |- ( w = ( J ` A ) -> suc w = suc ( J ` A ) )
24	23	fveq2d 6195	. . . . . . 7 |- ( w = ( J ` A ) -> ( F ` suc w ) = ( F ` suc ( J ` A ) ) )
25	22, 24	difeq12d 3729	. . . . . 6 |- ( w = ( J ` A ) -> ( ( F ` w ) \ ( F ` suc w ) ) = ( ( F ` ( J ` A ) ) \ ( F ` suc ( J ` A ) ) ) )
26		eqid 2622	. . . . . 6 |- ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) = ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) )
27		fvex 6201	. . . . . . 7 |- ( F ` ( J ` A ) ) e. _V
28		difexg 4808	. . . . . . 7 |- ( ( F ` ( J ` A ) ) e. _V -> ( ( F ` ( J ` A ) ) \ ( F ` suc ( J ` A ) ) ) e. _V )
29	27, 28	ax-mp 5	. . . . . 6 |- ( ( F ` ( J ` A ) ) \ ( F ` suc ( J ` A ) ) ) e. _V
30	25, 26, 29	fvmpt 6282	. . . . 5 |- ( ( J ` A ) e. S -> ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) ` ( J ` A ) ) = ( ( F ` ( J ` A ) ) \ ( F ` suc ( J ` A ) ) ) )
31	21, 30	syl 17	. . . 4 |- ( ( ph /\ A e. om ) -> ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) ` ( J ` A ) ) = ( ( F ` ( J ` A ) ) \ ( F ` suc ( J ` A ) ) ) )
32	16, 31	eqtrd 2656	. . 3 |- ( ( ph /\ A e. om ) -> ( ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) o. J ) ` A ) = ( ( F ` ( J ` A ) ) \ ( F ` suc ( J ` A ) ) ) )
33	2, 32	syl5eq 2668	. 2 |- ( ( ph /\ A e. om ) -> ( K ` A ) = ( ( F ` ( J ` A ) ) \ ( F ` suc ( J ` A ) ) ) )
34		suceq 5790	. . . . . . . . 9 |- ( y = ( J ` A ) -> suc y = suc ( J ` A ) )
35	34	fveq2d 6195	. . . . . . . 8 |- ( y = ( J ` A ) -> ( F ` suc y ) = ( F ` suc ( J ` A ) ) )
36		fveq2 6191	. . . . . . . 8 |- ( y = ( J ` A ) -> ( F ` y ) = ( F ` ( J ` A ) ) )
37	35, 36	psseq12d 3701	. . . . . . 7 |- ( y = ( J ` A ) -> ( ( F ` suc y ) C. ( F ` y ) <-> ( F ` suc ( J ` A ) ) C. ( F ` ( J ` A ) ) ) )
38	37, 3	elrab2 3366	. . . . . 6 |- ( ( J ` A ) e. S <-> ( ( J ` A ) e. om /\ ( F ` suc ( J ` A ) ) C. ( F ` ( J ` A ) ) ) )
39	38	simprbi 480	. . . . 5 |- ( ( J ` A ) e. S -> ( F ` suc ( J ` A ) ) C. ( F ` ( J ` A ) ) )
40	21, 39	syl 17	. . . 4 |- ( ( ph /\ A e. om ) -> ( F ` suc ( J ` A ) ) C. ( F ` ( J ` A ) ) )
41		df-pss 3590	. . . 4 |- ( ( F ` suc ( J ` A ) ) C. ( F ` ( J ` A ) ) <-> ( ( F ` suc ( J ` A ) ) C_ ( F ` ( J ` A ) ) /\ ( F ` suc ( J ` A ) ) =/= ( F ` ( J ` A ) ) ) )
42	40, 41	sylib 208	. . 3 |- ( ( ph /\ A e. om ) -> ( ( F ` suc ( J ` A ) ) C_ ( F ` ( J ` A ) ) /\ ( F ` suc ( J ` A ) ) =/= ( F ` ( J ` A ) ) ) )
43		pssdifn0 3944	. . 3 |- ( ( ( F ` suc ( J ` A ) ) C_ ( F ` ( J ` A ) ) /\ ( F ` suc ( J ` A ) ) =/= ( F ` ( J ` A ) ) ) -> ( ( F ` ( J ` A ) ) \ ( F ` suc ( J ` A ) ) ) =/= (/) )
44	42, 43	syl 17	. 2 |- ( ( ph /\ A e. om ) -> ( ( F ` ( J ` A ) ) \ ( F ` suc ( J ` A ) ) ) =/= (/) )
45	33, 44	eqnetrd 2861	1 |- ( ( ph /\ A e. om ) -> ( K ` A ) =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   o. ccom 5118   suc csuc 5725   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   iota_crio 6610   omcom 7065   ~~ cen 7952   Fincfn 7955

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-rep 4771  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-reu 2919  df-rmo 2920  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-se 5074  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-pred 5680  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  isf32lem9  9183