Theorem fin23lem31 8517

Description: Lemma for fin23 8563. 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 8504	. . 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 8515	. . . . 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 8513	. . . . . . 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 3651	. . . . . 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 6915	. . . . . . . . . . . . . 14 |- U Fn om
15		fndm 5515	. . . . . . . . . . . . . 14 |- ( U Fn om -> dom U = om )
16	14, 15	ax-mp 5	. . . . . . . . . . . . 13 |- dom U = om
17		peano1 6500	. . . . . . . . . . . . . 14 |- (/) e. om
18		ne0i 3648	. . . . . . . . . . . . . 14 |- ( (/) e. om -> om =/= (/) )
19	17, 18	ax-mp 5	. . . . . . . . . . . . 13 |- om =/= (/)
20	16, 19	eqnetri 2630	. . . . . . . . . . . 12 |- dom U =/= (/)
21		dm0rn0 5061	. . . . . . . . . . . . 13 |- ( dom U = (/) <-> ran U = (/) )
22	21	necon3bii 2645	. . . . . . . . . . . 12 |- ( dom U =/= (/) <-> ran U =/= (/) )
23	20, 22	mpbi 208	. . . . . . . . . . 11 |- ran U =/= (/)
24		intssuni 4155	. . . . . . . . . . 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 8509	. . . . . . . . . 10 |- U. ran U = U. ran t
27	25, 26	sseqtri 3393	. . . . . . . . 9 |- |^| ran U C_ U. ran t
28	27	sseli 3357	. . . . . . . 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 5613	. . . . . . . . . . . . 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 8516	. . . . . . . . . . . 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 3724	. . . . . . . . . . 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 2781	. . . . . . . . . 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 2706	. . . . . . 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 2700	. . . . 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 1692	. . . 4 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> U. ran Z =/= U. ran t )
44		df-pss 3349	. . . 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 1183	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 965   = wceq 1369   E.wex 1586   e. wcel 1756   {cab 2429   =/= wne 2611   A.wral 2720   {crab 2724   _Vcvv 2977   \ cdif 3330   i^i cin 3332   C_ wss 3333   C. wpss 3334   (/)c0 3642   ifcif 3796   ~Pcpw 3865   U.cuni 4096   |^|cint 4133   class class class wbr 4297   e. cmpt 4355   suc csuc 4726   dom cdm 4845   ran crn 4846   o. ccom 4849   Fun wfun 5417   Fn wfn 5418   -1-1->wf1 5420   ` cfv 5423   iota_crio 6056  (class class class)co 6096   e. cmpt2 6098   omcom 6481  seq_𝜔cseqom 6907   ^m cmap 7219   ~~ cen 7312   Fincfn 7315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-seqom 6908  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114

This theorem is referenced by:  fin23lem32  8518