Theorem fin23lem31 8719

Description: Lemma for fin23 8765. 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 8706	. . 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 8717	. . . . 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 8715	. . . . . . 7 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> |^| ran U =/= (/) )
11	10	ancoms 453	. . . . . 6 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> |^| ran U =/= (/) )
12		n0 3794	. . . . . 6 |- ( |^| ran U =/= (/) <-> E. a a e. |^| ran U )
13	11, 12	sylib 196	. . . . 5 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> E. a a e. |^| ran U )
14	3	fnseqom 7117	. . . . . . . . . . . . . 14 |- U Fn om
15		fndm 5678	. . . . . . . . . . . . . 14 |- ( U Fn om -> dom U = om )
16	14, 15	ax-mp 5	. . . . . . . . . . . . 13 |- dom U = om
17		peano1 6697	. . . . . . . . . . . . . 14 |- (/) e. om
18		ne0i 3791	. . . . . . . . . . . . . 14 |- ( (/) e. om -> om =/= (/) )
19	17, 18	ax-mp 5	. . . . . . . . . . . . 13 |- om =/= (/)
20	16, 19	eqnetri 2763	. . . . . . . . . . . 12 |- dom U =/= (/)
21		dm0rn0 5217	. . . . . . . . . . . . 13 |- ( dom U = (/) <-> ran U = (/) )
22	21	necon3bii 2735	. . . . . . . . . . . 12 |- ( dom U =/= (/) <-> ran U =/= (/) )
23	20, 22	mpbi 208	. . . . . . . . . . 11 |- ran U =/= (/)
24		intssuni 4304	. . . . . . . . . . 11 |- ( ran U =/= (/) -> |^| ran U C_ U. ran U )
25	23, 24	ax-mp 5	. . . . . . . . . 10 |- |^| ran U C_ U. ran U
26	3	fin23lem16 8711	. . . . . . . . . 10 |- U. ran U = U. ran t
27	25, 26	sseqtri 3536	. . . . . . . . 9 |- |^| ran U C_ U. ran t
28	27	sseli 3500	. . . . . . . 8 |- ( a e. |^| ran U -> a e. U. ran t )
29	28	adantl 466	. . . . . . 7 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> a e. U. ran t )
30		f1fun 5781	. . . . . . . . . . . . 13 |- ( t : om -1-1-> V -> Fun t )
31	30	adantr 465	. . . . . . . . . . . 12 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> Fun t )
32	3, 1, 4, 5, 6, 7	fin23lem30 8718	. . . . . . . . . . . 12 |- ( Fun t -> ( U. ran Z i^i |^| ran U ) = (/) )
33	31, 32	syl 16	. . . . . . . . . . 11 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> ( U. ran Z i^i |^| ran U ) = (/) )
34		disj 3867	. . . . . . . . . . 11 |- ( ( U. ran Z i^i |^| ran U ) = (/) <-> A. a e. U. ran Z -. a e. |^| ran U )
35	33, 34	sylib 196	. . . . . . . . . 10 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> A. a e. U. ran Z -. a e. |^| ran U )
36		rsp 2830	. . . . . . . . . 10 |- ( A. a e. U. ran Z -. a e. |^| ran U -> ( a e. U. ran Z -> -. a e. |^| ran U ) )
37	35, 36	syl 16	. . . . . . . . 9 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> ( a e. U. ran Z -> -. a e. |^| ran U ) )
38	37	con2d 115	. . . . . . . 8 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> ( a e. |^| ran U -> -. a e. U. ran Z ) )
39	38	imp 429	. . . . . . 7 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> -. a e. U. ran Z )
40		nelne1 2796	. . . . . . 7 |- ( ( a e. U. ran t /\ -. a e. U. ran Z ) -> U. ran t =/= U. ran Z )
41	29, 39, 40	syl2anc 661	. . . . . 6 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> U. ran t =/= U. ran Z )
42	41	necomd 2738	. . . . 5 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> U. ran Z =/= U. ran t )
43	13, 42	exlimddv 1702	. . . 4 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> U. ran Z =/= U. ran t )
44		df-pss 3492	. . . 4 |- ( U. ran Z C. U. ran t <-> ( U. ran Z C_ U. ran t /\ U. ran Z =/= U. ran t ) )
45	9, 43, 44	sylanbrc 664	. . 3 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> U. ran Z C. U. ran t )
46	2, 45	sylan2 474	. 2 |- ( ( t : om -1-1-> V /\ ( G e. F /\ U. ran t C_ G ) ) -> U. ran Z C. U. ran t )
47	46	3impb 1192	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 369   /\ w3a 973   = wceq 1379   E.wex 1596   e. wcel 1767   {cab 2452   =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113   \ cdif 3473   i^i cin 3475   C_ wss 3476   C. wpss 3477   (/)c0 3785   ifcif 3939   ~Pcpw 4010   U.cuni 4245   |^|cint 4282   class class class wbr 4447   |-> cmpt 4505   suc csuc 4880   dom cdm 4999   ran crn 5000   o. ccom 5003   Fun wfun 5580   Fn wfn 5581   -1-1->wf1 5583   ` cfv 5586   iota_crio 6242  (class class class)co 6282   |-> cmpt2 6284   omcom 6678  seq_𝜔cseqom 7109   ^m cmap 7417   ~~ cen 7510   Fincfn 7513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-seqom 7110  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316

This theorem is referenced by:  fin23lem32  8720