Theorem fin23lem31 8770

Description: Lemma for fin23 8816. 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 8757	. . 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 8768	. . . . 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 8766	. . . . . . 7 |- ( ( U. ran t e. F /\ t : om -1-1-> V ) -> |^| ran U =/= (/) )
11	10	ancoms 455	. . . . . 6 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> |^| ran U =/= (/) )
12		n0 3740	. . . . . 6 |- ( |^| ran U =/= (/) <-> E. a a e. |^| ran U )
13	11, 12	sylib 200	. . . . 5 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> E. a a e. |^| ran U )
14	3	fnseqom 7169	. . . . . . . . . . . . . 14 |- U Fn om
15		fndm 5673	. . . . . . . . . . . . . 14 |- ( U Fn om -> dom U = om )
16	14, 15	ax-mp 5	. . . . . . . . . . . . 13 |- dom U = om
17		peano1 6709	. . . . . . . . . . . . . 14 |- (/) e. om
18	17	ne0ii 3737	. . . . . . . . . . . . 13 |- om =/= (/)
19	16, 18	eqnetri 2693	. . . . . . . . . . . 12 |- dom U =/= (/)
20		dm0rn0 5050	. . . . . . . . . . . . 13 |- ( dom U = (/) <-> ran U = (/) )
21	20	necon3bii 2675	. . . . . . . . . . . 12 |- ( dom U =/= (/) <-> ran U =/= (/) )
22	19, 21	mpbi 212	. . . . . . . . . . 11 |- ran U =/= (/)
23		intssuni 4256	. . . . . . . . . . 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 8762	. . . . . . . . . 10 |- U. ran U = U. ran t
26	24, 25	sseqtri 3463	. . . . . . . . 9 |- |^| ran U C_ U. ran t
27	26	sseli 3427	. . . . . . . 8 |- ( a e. |^| ran U -> a e. U. ran t )
28	27	adantl 468	. . . . . . 7 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> a e. U. ran t )
29		f1fun 5779	. . . . . . . . . . . . 13 |- ( t : om -1-1-> V -> Fun t )
30	29	adantr 467	. . . . . . . . . . . 12 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> Fun t )
31	3, 1, 4, 5, 6, 7	fin23lem30 8769	. . . . . . . . . . . 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 3804	. . . . . . . . . . 11 |- ( ( U. ran Z i^i |^| ran U ) = (/) <-> A. a e. U. ran Z -. a e. |^| ran U )
34	32, 33	sylib 200	. . . . . . . . . 10 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> A. a e. U. ran Z -. a e. |^| ran U )
35		rsp 2753	. . . . . . . . . 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 119	. . . . . . . 8 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> ( a e. |^| ran U -> -. a e. U. ran Z ) )
38	37	imp 431	. . . . . . 7 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> -. a e. U. ran Z )
39		nelne1 2719	. . . . . . 7 |- ( ( a e. U. ran t /\ -. a e. U. ran Z ) -> U. ran t =/= U. ran Z )
40	28, 38, 39	syl2anc 666	. . . . . 6 |- ( ( ( t : om -1-1-> V /\ U. ran t e. F ) /\ a e. |^| ran U ) -> U. ran t =/= U. ran Z )
41	40	necomd 2678	. . . . 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 1780	. . . 4 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> U. ran Z =/= U. ran t )
43		df-pss 3419	. . . 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 669	. . 3 |- ( ( t : om -1-1-> V /\ U. ran t e. F ) -> U. ran Z C. U. ran t )
45	2, 44	sylan2 477	. 2 |- ( ( t : om -1-1-> V /\ ( G e. F /\ U. ran t C_ G ) ) -> U. ran Z C. U. ran t )
46	45	3impb 1203	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 371   /\ w3a 984   = wceq 1443   E.wex 1662   e. wcel 1886   {cab 2436   =/= wne 2621   A.wral 2736   {crab 2740   _Vcvv 3044   \ cdif 3400   i^i cin 3402   C_ wss 3403   C. wpss 3404   (/)c0 3730   ifcif 3880   ~Pcpw 3950   U.cuni 4197   |^|cint 4233   class class class wbr 4401   |-> cmpt 4460   dom cdm 4833   ran crn 4834   o. ccom 4837   suc csuc 5424   Fun wfun 5575   Fn wfn 5576   -1-1->wf1 5578   ` cfv 5581   iota_crio 6249  (class class class)co 6288   |-> cmpt2 6290   omcom 6689  seq_𝜔cseqom 7161   ^m cmap 7469   ~~ cen 7563   Fincfn 7566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-seqom 7162  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370

This theorem is referenced by:  fin23lem32  8771