Theorem fin23lem31 9165

Description: Lemma for fin23 9211. The residual is has a strictly smaller range than the previous sequence. This will be iterated to build an unbounded chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Hypotheses

Ref	Expression
fin23lem.a	|- U = seq𝜔 ( ( i e. om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) ) , U. ran t )
fin23lem17.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 ) }
fin23lem.b	|- P = { v e. om | |^| ran U C_ ( t ` v ) }
fin23lem.c	|- Q = ( w e. om |-> ( iota_ x e. P ( x i^i P ) ~~ w ) )
fin23lem.d	|- R = ( w e. om |-> ( iota_ x e. ( om \ P ) ( x i^i ( om \ P ) ) ~~ w ) )
fin23lem.e	|- Z = if ( P e. Fin , ( t o. R ) , ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) )

Assertion

Ref	Expression
fin23lem31	|- ( ( t : om -1-1-> V /\ G e. F /\ U. ran t C_ G ) -> U. ran Z C. U. ran t )

Distinct variable groups:   g, i, t, u, v, x, z, a   F, a, t   V, a   w, a, x, z, P   v, a, R, i, u   U, a, i, u, v, z   Z, a   g, a, G, t, x

Allowed substitution hints:   P( v, u, t, g, i)   Q( x, z, w, v, u, t, g, i, a)   R( x, z, w, t, g)   U( x, w, t, g)   F( x, z, w, v, u, g, i)   G( z, w, v, u, i)   V( x, z, w, v, u, t, g, i)   Z( x, z, w, v, u, t, g, i)

Proof of Theorem fin23lem31

Step	Hyp	Ref	Expression
1		fin23lem17.f	. . . 4 |- F = { g | A. a e. ( ~P g ^m om ) ( A. x e. om ( a ` suc x ) C_ ( a ` x ) -> |^| ran a e. ran a ) }
2	1	ssfin3ds 9152	. . 3 |- ( ( G e. F /\ U. ran t C_ G ) -> U. ran t e. F )
3		fin23lem.a	. . . . . 6 |- U = seq𝜔 ( ( i e. om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) ) , U. ran t )
4		fin23lem.b	. . . . . 6 |- P = { v e. om | |^| ran U C_ ( t ` v ) }
5		fin23lem.c	. . . . . 6 |- Q = ( w e. om |-> ( iota_ x e. P ( x i^i P ) ~~ w ) )
6		fin23lem.d	. . . . . 6 |- R = ( w e. om |-> ( iota_ x e. ( om \ P ) ( x i^i ( om \ P ) ) ~~ w ) )
7		fin23lem.e	. . . . . 6 |- Z = if ( P e. Fin , ( t o. R ) , ( ( z e. P |-> ( ( t ` z ) \ |^| ran U ) ) o. Q ) )
8	3, 1, 4, 5, 6, 7	fin23lem29 9163	. . . . 5 |- U. ran Z C_ U. ran t
9	8	a1i 11	. . . 4 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> U. ran Z C_ U. ran t )
10	3, 1	fin23lem21 9161	. . . . . . 7 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> |^| ran U =/= (/) )
11	10	ancoms 469	. . . . . 6 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> |^| ran U =/= (/) )
12		n0 3931	. . . . . 6 |- ( |^| ran U =/= (/) <-> E. a a e. |^| ran U )
13	11, 12	sylib 208	. . . . 5 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> E. a a e. |^| ran U )
14	3	fnseqom 7550	. . . . . . . . . . . . . 14 |- U Fn om
15		fndm 5990	. . . . . . . . . . . . . 14 |- ( U Fn om -> dom U = om )
16	14, 15	ax-mp 5	. . . . . . . . . . . . 13 |- dom U = om
17		peano1 7085	. . . . . . . . . . . . . 14 |- (/) e. om
18	17	ne0ii 3923	. . . . . . . . . . . . 13 |- om =/= (/)
19	16, 18	eqnetri 2864	. . . . . . . . . . . 12 |- dom U =/= (/)
20		dm0rn0 5342	. . . . . . . . . . . . 13 |- ( dom U = (/) <-> ran U = (/) )
21	20	necon3bii 2846	. . . . . . . . . . . 12 |- ( dom U =/= (/) <-> ran U =/= (/) )
22	19, 21	mpbi 220	. . . . . . . . . . 11 |- ran U =/= (/)
23		intssuni 4499	. . . . . . . . . . 11 |- ( ran U =/= (/) -> |^| ran U C_ U. ran U )
24	22, 23	ax-mp 5	. . . . . . . . . 10 |- |^| ran U C_ U. ran U
25	3	fin23lem16 9157	. . . . . . . . . 10 |- U. ran U = U. ran t
26	24, 25	sseqtri 3637	. . . . . . . . 9 |- |^| ran U C_ U. ran t
27	26	sseli 3599	. . . . . . . 8 |- ( a e. |^| ran U -> a e. U. ran t )
28	27	adantl 482	. . . . . . 7 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> a e. U. ran t )
29		f1fun 6103	. . . . . . . . . . . . 13 |- ( t : om -1-1-> V -> Fun t )
30	29	adantr 481	. . . . . . . . . . . 12 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> Fun t )
31	3, 1, 4, 5, 6, 7	fin23lem30 9164	. . . . . . . . . . . 12 |- ( Fun t -> ( U. ran Z i^i |^| ran U ) = (/) )
32	30, 31	syl 17	. . . . . . . . . . 11 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> ( U. ran Z i^i |^| ran U ) = (/) )
33		disj 4017	. . . . . . . . . . 11 |- ( ( U. ran Z i^i |^| ran U ) = (/) <-> A. a e. U. ran Z -. a e. |^| ran U )
34	32, 33	sylib 208	. . . . . . . . . 10 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> A. a e. U. ran Z -. a e. |^| ran U )
35		rsp 2929	. . . . . . . . . 10 |- ( A. a e. U. ran Z -. a e. |^| ran U -> ( a e. U. ran Z -> -. a e. |^| ran U ) )
36	34, 35	syl 17	. . . . . . . . 9 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> ( a e. U. ran Z -> -. a e. |^| ran U ) )
37	36	con2d 129	. . . . . . . 8 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> ( a e. |^| ran U -> -. a e. U. ran Z ) )
38	37	imp 445	. . . . . . 7 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> -. a e. U. ran Z )
39		nelne1 2890	. . . . . . 7 |- ( ( a e. U. ran t /\ -. a e. U. ran Z ) -> U. ran t =/= U. ran Z )
40	28, 38, 39	syl2anc 693	. . . . . 6 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> U. ran t =/= U. ran Z )
41	40	necomd 2849	. . . . 5 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> U. ran Z =/= U. ran t )
42	13, 41	exlimddv 1863	. . . 4 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> U. ran Z =/= U. ran t )
43		df-pss 3590	. . . 4 |- ( U. ran Z C. U. ran t <-> ( U. ran Z C_ U. ran t /\ U. ran Z =/= U. ran t ) )
44	9, 42, 43	sylanbrc 698	. . 3 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> U. ran Z C. U. ran t )
45	2, 44	sylan2 491	. 2 |- ( ( t : om -1-1-> V /\ ( G e. F /\ U. ran t C_ G ) ) -> U. ran Z C. U. ran t )
46	45	3impb 1260	1 |- ( ( t : om -1-1-> V /\ G e. F /\ U. ran t C_ G ) -> U. ran Z C. U. ran t )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)c0 3915   ifcif 4086   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   o. ccom 5118   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   -1-1->wf1 5885   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   omcom 7065  seq_𝜔cseqom 7542   ^m cmap 7857   ~~ 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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  fin23lem32  9166