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Theorem unblem2 7764
Description: Lemma for unbnn 7767. 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.
StepHypRef Expression
1 fveq2 5859 . . . 4  |-  ( z  =  (/)  ->  ( F `
 z )  =  ( F `  (/) ) )
21eleq1d 2531 . . 3  |-  ( z  =  (/)  ->  ( ( F `  z )  e.  A  <->  ( F `  (/) )  e.  A
) )
3 fveq2 5859 . . . 4  |-  ( z  =  u  ->  ( F `  z )  =  ( F `  u ) )
43eleq1d 2531 . . 3  |-  ( z  =  u  ->  (
( F `  z
)  e.  A  <->  ( F `  u )  e.  A
) )
5 fveq2 5859 . . . 4  |-  ( z  =  suc  u  -> 
( F `  z
)  =  ( F `
 suc  u )
)
65eleq1d 2531 . . 3  |-  ( z  =  suc  u  -> 
( ( F `  z )  e.  A  <->  ( F `  suc  u
)  e.  A ) )
7 omsson 6677 . . . . . 6  |-  om  C_  On
8 sstr 3507 . . . . . 6  |-  ( ( A  C_  om  /\  om  C_  On )  ->  A  C_  On )
97, 8mpan2 671 . . . . 5  |-  ( A 
C_  om  ->  A  C_  On )
10 peano1 6692 . . . . . . . . 9  |-  (/)  e.  om
11 eleq1 2534 . . . . . . . . . . 11  |-  ( w  =  (/)  ->  ( w  e.  v  <->  (/)  e.  v ) )
1211rexbidv 2968 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( E. v  e.  A  w  e.  v  <->  E. v  e.  A  (/)  e.  v ) )
1312rspcv 3205 . . . . . . . . 9  |-  ( (/)  e.  om  ->  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  E. v  e.  A  (/) 
e.  v ) )
1410, 13ax-mp 5 . . . . . . . 8  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  E. v  e.  A  (/) 
e.  v )
15 df-rex 2815 . . . . . . . 8  |-  ( E. v  e.  A  (/)  e.  v  <->  E. v ( v  e.  A  /\  (/)  e.  v ) )
1614, 15sylib 196 . . . . . . 7  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  E. v ( v  e.  A  /\  (/)  e.  v ) )
17 exsimpl 1649 . . . . . . 7  |-  ( E. v ( v  e.  A  /\  (/)  e.  v )  ->  E. v 
v  e.  A )
1816, 17syl 16 . . . . . 6  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  E. v  v  e.  A )
19 n0 3789 . . . . . 6  |-  ( A  =/=  (/)  <->  E. v  v  e.  A )
2018, 19sylibr 212 . . . . 5  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  A  =/=  (/) )
21 onint 6603 . . . . 5  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
229, 20, 21syl2an 477 . . . 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 )
2423fveq1i 5860 . . . . . . 7  |-  ( F `
 (/) )  =  ( ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om ) `  (/) )
25 fr0g 7093 . . . . . . 7  |-  ( |^| A  e.  A  ->  ( ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om ) `  (/) )  = 
|^| A )
2624, 25syl5req 2516 . . . . . 6  |-  ( |^| A  e.  A  ->  |^| A  =  ( F `
 (/) ) )
2726eleq1d 2531 . . . . 5  |-  ( |^| A  e.  A  ->  (
|^| A  e.  A  <->  ( F `  (/) )  e.  A ) )
2827ibi 241 . . . 4  |-  ( |^| A  e.  A  ->  ( F `  (/) )  e.  A )
2922, 28syl 16 . . 3  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  ( F `  (/) )  e.  A )
30 unblem1 7763 . . . . 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 4938 . . . . . . . . . . . 12  |-  ( y  =  x  ->  suc  y  =  suc  x )
3231difeq2d 3617 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( A  \  suc  y )  =  ( A  \  suc  x ) )
3332inteqd 4282 . . . . . . . . . 10  |-  ( y  =  x  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  x )
)
34 suceq 4938 . . . . . . . . . . . 12  |-  ( y  =  ( F `  u )  ->  suc  y  =  suc  ( F `
 u ) )
3534difeq2d 3617 . . . . . . . . . . 11  |-  ( y  =  ( F `  u )  ->  ( A  \  suc  y )  =  ( A  \  suc  ( F `  u
) ) )
3635inteqd 4282 . . . . . . . . . 10  |-  ( y  =  ( F `  u )  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  ( F `  u ) ) )
3723, 33, 36frsucmpt2 7097 . . . . . . . . 9  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  ( F `  suc  u )  =  |^| ( A  \  suc  ( F `  u )
) )
3837eqcomd 2470 . . . . . . . 8  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  |^| ( A  \  suc  ( F `  u
) )  =  ( F `  suc  u
) )
3938eleq1d 2531 . . . . . . 7  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  ( |^| ( A  \  suc  ( F `
 u ) )  e.  A  <->  ( F `  suc  u )  e.  A ) )
4039ex 434 . . . . . 6  |-  ( u  e.  om  ->  ( |^| ( A  \  suc  ( F `  u ) )  e.  A  -> 
( |^| ( A  \  suc  ( F `  u
) )  e.  A  <->  ( F `  suc  u
)  e.  A ) ) )
4140ibd 243 . . . . 5  |-  ( u  e.  om  ->  ( |^| ( A  \  suc  ( F `  u ) )  e.  A  -> 
( F `  suc  u )  e.  A
) )
4230, 41syl5 32 . . . 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
) )
4342expd 436 . . 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
) ) )
442, 4, 6, 29, 43finds2 6701 . 2  |-  ( z  e.  om  ->  (
( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  ( F `  z )  e.  A ) )
4544com12 31 1  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
z  e.  om  ->  ( F `  z )  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2657   A.wral 2809   E.wrex 2810   _Vcvv 3108    \ cdif 3468    C_ wss 3471   (/)c0 3780   |^|cint 4277    |-> cmpt 4500   Oncon0 4873   suc csuc 4875    |` cres 4996   ` cfv 5581   omcom 6673   reccrdg 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6674  df-recs 7034  df-rdg 7068
This theorem is referenced by:  unblem3  7765  unblem4  7766
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