Theorem unblem2 8213

Description: Lemma for unbnn 8216. The value of the function F belongs to the unbounded set of natural numbers A. (Contributed by NM, 3-Dec-2003.)

Hypothesis

Ref	Expression
unblem.2	|- F = ( rec ( ( x e. _V |-> |^| ( A \ suc x ) ) , |^| A ) |` om )

Assertion

Ref	Expression
unblem2	|- ( ( A C_ om /\ A. w e. om E. v e. A w e. v ) -> ( z e. om -> ( F ` z ) e. A ) )

Distinct variable groups:   w, v, x, z, A   v, F, w, z

Allowed substitution hint:   F( x)

Proof of Theorem unblem2

Dummy variables u y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( z = (/) -> ( F ` z ) = ( F ` (/) ) )
2	1	eleq1d 2686	. . 3 |- ( z = (/) -> ( ( F ` z ) e. A <-> ( F ` (/) ) e. A ) )
3		fveq2 6191	. . . 4 |- ( z = u -> ( F ` z ) = ( F ` u ) )
4	3	eleq1d 2686	. . 3 |- ( z = u -> ( ( F ` z ) e. A <-> ( F ` u ) e. A ) )
5		fveq2 6191	. . . 4 |- ( z = suc u -> ( F ` z ) = ( F ` suc u ) )
6	5	eleq1d 2686	. . 3 |- ( z = suc u -> ( ( F ` z ) e. A <-> ( F ` suc u ) e. A ) )
7		omsson 7069	. . . . . 6 |- om C_ On
8		sstr 3611	. . . . . 6 |- ( ( A C_ om /\ om C_ On ) -> A C_ On )
9	7, 8	mpan2 707	. . . . 5 |- ( A C_ om -> A C_ On )
10		peano1 7085	. . . . . . . . 9 |- (/) e. om
11		eleq1 2689	. . . . . . . . . . 11 |- ( w = (/) -> ( w e. v <-> (/) e. v ) )
12	11	rexbidv 3052	. . . . . . . . . 10 |- ( w = (/) -> ( E. v e. A w e. v <-> E. v e. A (/) e. v ) )
13	12	rspcv 3305	. . . . . . . . 9 |- ( (/) e. om -> ( A. w e. om E. v e. A w e. v -> E. v e. A (/) e. v ) )
14	10, 13	ax-mp 5	. . . . . . . 8 |- ( A. w e. om E. v e. A w e. v -> E. v e. A (/) e. v )
15		df-rex 2918	. . . . . . . 8 |- ( E. v e. A (/) e. v <-> E. v ( v e. A /\ (/) e. v ) )
16	14, 15	sylib 208	. . . . . . 7 |- ( A. w e. om E. v e. A w e. v -> E. v ( v e. A /\ (/) e. v ) )
17		exsimpl 1795	. . . . . . 7 |- ( E. v ( v e. A /\ (/) e. v ) -> E. v v e. A )
18	16, 17	syl 17	. . . . . 6 |- ( A. w e. om E. v e. A w e. v -> E. v v e. A )
19		n0 3931	. . . . . 6 |- ( A =/= (/) <-> E. v v e. A )
20	18, 19	sylibr 224	. . . . 5 |- ( A. w e. om E. v e. A w e. v -> A =/= (/) )
21		onint 6995	. . . . 5 |- ( ( A C_ On /\ A =/= (/) ) -> |^| A e. A )
22	9, 20, 21	syl2an 494	. . . 4 |- ( ( A C_ om /\ A. w e. om E. v e. A w e. v ) -> |^| A e. A )
23		unblem.2	. . . . . . . 8 |- F = ( rec ( ( x e. _V |-> |^| ( A \ suc x ) ) , |^| A ) |` om )
24	23	fveq1i 6192	. . . . . . 7 |- ( F ` (/) ) = ( ( rec ( ( x e. _V |-> |^| ( A \ suc x ) ) , |^| A ) |` om ) ` (/) )
25		fr0g 7531	. . . . . . 7 |- ( |^| A e. A -> ( ( rec ( ( x e. _V |-> |^| ( A \ suc x ) ) , |^| A ) |` om ) ` (/) ) = |^| A )
26	24, 25	syl5req 2669	. . . . . 6 |- ( |^| A e. A -> |^| A = ( F ` (/) ) )
27	26	eleq1d 2686	. . . . 5 |- ( |^| A e. A -> ( |^| A e. A <-> ( F ` (/) ) e. A ) )
28	27	ibi 256	. . . 4 |- ( |^| A e. A -> ( F ` (/) ) e. A )
29	22, 28	syl 17	. . 3 |- ( ( A C_ om /\ A. w e. om E. v e. A w e. v ) -> ( F ` (/) ) e. A )
30		unblem1 8212	. . . . 5 |- ( ( ( A C_ om /\ A. w e. om E. v e. A w e. v ) /\ ( F ` u ) e. A ) -> |^| ( A \ suc ( F ` u ) ) e. A )
31		suceq 5790	. . . . . . . . . . . 12 |- ( y = x -> suc y = suc x )
32	31	difeq2d 3728	. . . . . . . . . . 11 |- ( y = x -> ( A \ suc y ) = ( A \ suc x ) )
33	32	inteqd 4480	. . . . . . . . . 10 |- ( y = x -> |^| ( A \ suc y ) = |^| ( A \ suc x ) )
34		suceq 5790	. . . . . . . . . . . 12 |- ( y = ( F ` u ) -> suc y = suc ( F ` u ) )
35	34	difeq2d 3728	. . . . . . . . . . 11 |- ( y = ( F ` u ) -> ( A \ suc y ) = ( A \ suc ( F ` u ) ) )
36	35	inteqd 4480	. . . . . . . . . 10 |- ( y = ( F ` u ) -> |^| ( A \ suc y ) = |^| ( A \ suc ( F ` u ) ) )
37	23, 33, 36	frsucmpt2 7535	. . . . . . . . 9 |- ( ( u e. om /\ |^| ( A \ suc ( F ` u ) ) e. A ) -> ( F ` suc u ) = |^| ( A \ suc ( F ` u ) ) )
38	37	eqcomd 2628	. . . . . . . 8 |- ( ( u e. om /\ |^| ( A \ suc ( F ` u ) ) e. A ) -> |^| ( A \ suc ( F ` u ) ) = ( F ` suc u ) )
39	38	eleq1d 2686	. . . . . . 7 |- ( ( u e. om /\ |^| ( A \ suc ( F ` u ) ) e. A ) -> ( |^| ( A \ suc ( F ` u ) ) e. A <-> ( F ` suc u ) e. A ) )
40	39	ex 450	. . . . . 6 |- ( u e. om -> ( |^| ( A \ suc ( F ` u ) ) e. A -> ( |^| ( A \ suc ( F ` u ) ) e. A <-> ( F ` suc u ) e. A ) ) )
41	40	ibd 258	. . . . 5 |- ( u e. om -> ( |^| ( A \ suc ( F ` u ) ) e. A -> ( F ` suc u ) e. A ) )
42	30, 41	syl5 34	. . . 4 |- ( u e. om -> ( ( ( A C_ om /\ A. w e. om E. v e. A w e. v ) /\ ( F ` u ) e. A ) -> ( F ` suc u ) e. A ) )
43	42	expd 452	. . 3 |- ( u e. om -> ( ( A C_ om /\ A. w e. om E. v e. A w e. v ) -> ( ( F ` u ) e. A -> ( F ` suc u ) e. A ) ) )
44	2, 4, 6, 29, 43	finds2 7094	. 2 |- ( z e. om -> ( ( A C_ om /\ A. w e. om E. v e. A w e. v ) -> ( F ` z ) e. A ) )
45	44	com12 32	1 |- ( ( A C_ om /\ A. w e. om E. v e. A w e. v ) -> ( z e. om -> ( F ` z ) e. A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   |^|cint 4475   |-> cmpt 4729   |` cres 5116   Oncon0 5723   suc csuc 5725   ` cfv 5888   omcom 7065   reccrdg 7505

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-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-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-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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  unblem3  8214  unblem4  8215