Theorem fin23lem31 8740

Description: Lemma for fin23 8786. 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 8727	. . 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 8738	. . . . 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 8736	. . . . . . 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 3803	. . . . . 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 7138	. . . . . . . . . . . . . 14 |- U Fn om
15		fndm 5686	. . . . . . . . . . . . . 14 |- ( U Fn om -> dom U = om )
16	14, 15	ax-mp 5	. . . . . . . . . . . . 13 |- dom U = om
17		peano1 6718	. . . . . . . . . . . . . 14 |- (/) e. om
18	17	ne0ii 3800	. . . . . . . . . . . . 13 |- om =/= (/)
19	16, 18	eqnetri 2753	. . . . . . . . . . . 12 |- dom U =/= (/)
20		dm0rn0 5229	. . . . . . . . . . . . 13 |- ( dom U = (/) <-> ran U = (/) )
21	20	necon3bii 2725	. . . . . . . . . . . 12 |- ( dom U =/= (/) <-> ran U =/= (/) )
22	19, 21	mpbi 208	. . . . . . . . . . 11 |- ran U =/= (/)
23		intssuni 4311	. . . . . . . . . . 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 8732	. . . . . . . . . 10 |- U. ran U = U. ran t
26	24, 25	sseqtri 3531	. . . . . . . . 9 |- |^| ran U C_ U. ran t
27	26	sseli 3495	. . . . . . . 8 |- ( a e. |^| ran U -> a e. U. ran t )
28	27	adantl 466	. . . . . . 7 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> a e. U. ran t )
29		f1fun 5789	. . . . . . . . . . . . 13 |- ( t : om -1-1-> V -> Fun t )
30	29	adantr 465	. . . . . . . . . . . 12 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> Fun t )
31	3, 1, 4, 5, 6, 7	fin23lem30 8739	. . . . . . . . . . . 12 |- ( Fun t -> ( U. ran Z i^i |^| ran U ) = (/) )
32	30, 31	syl 16	. . . . . . . . . . 11 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> ( U. ran Z i^i |^| ran U ) = (/) )
33		disj 3870	. . . . . . . . . . 11 |- ( ( U. ran Z i^i |^| ran U ) = (/) <-> A. a e. U. ran Z -. a e. |^| ran U )
34	32, 33	sylib 196	. . . . . . . . . 10 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> A. a e. U. ran Z -. a e. |^| ran U )
35		rsp 2823	. . . . . . . . . 10 |- ( A. a e. U. ran Z -. a e. |^| ran U -> ( a e. U. ran Z -> -. a e. |^| ran U ) )
36	34, 35	syl 16	. . . . . . . . 9 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> ( a e. U. ran Z -> -. a e. |^| ran U ) )
37	36	con2d 115	. . . . . . . 8 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> ( a e. |^| ran U -> -. a e. U. ran Z ) )
38	37	imp 429	. . . . . . 7 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> -. a e. U. ran Z )
39		nelne1 2786	. . . . . . 7 |- ( ( a e. U. ran t /\ -. a e. U. ran Z ) -> U. ran t =/= U. ran Z )
40	28, 38, 39	syl2anc 661	. . . . . 6 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> U. ran t =/= U. ran Z )
41	40	necomd 2728	. . . . 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 1727	. . . 4 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> U. ran Z =/= U. ran t )
43		df-pss 3487	. . . 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 664	. . 3 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> U. ran Z C. U. ran t )
45	2, 44	sylan2 474	. 2 |- ( ( t : om -1-1-> V /\ ( G e. F /\ U. ran t C_ G ) ) -> U. ran Z C. U. ran t )
46	45	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 1395   E.wex 1613   e. wcel 1819   {cab 2442   =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109   \ cdif 3468   i^i cin 3470   C_ wss 3471   C. wpss 3472   (/)c0 3793   ifcif 3944   ~Pcpw 4015   U.cuni 4251   |^|cint 4288   class class class wbr 4456   |-> cmpt 4515   suc csuc 4889   dom cdm 5008   ran crn 5009   o. ccom 5012   Fun wfun 5588   Fn wfn 5589   -1-1->wf1 5591   ` cfv 5594   iota_crio 6257  (class class class)co 6296   |-> cmpt2 6298   omcom 6699  seq_𝜔cseqom 7130   ^m cmap 7438   ~~ cen 7532   Fincfn 7535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-seqom 7131  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337

This theorem is referenced by:  fin23lem32  8741