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Theorem unblem2 7360
Description: Lemma for unbnn 7363. 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 5728 . . . 4  |-  ( z  =  (/)  ->  ( F `
 z )  =  ( F `  (/) ) )
21eleq1d 2502 . . 3  |-  ( z  =  (/)  ->  ( ( F `  z )  e.  A  <->  ( F `  (/) )  e.  A
) )
3 fveq2 5728 . . . 4  |-  ( z  =  u  ->  ( F `  z )  =  ( F `  u ) )
43eleq1d 2502 . . 3  |-  ( z  =  u  ->  (
( F `  z
)  e.  A  <->  ( F `  u )  e.  A
) )
5 fveq2 5728 . . . 4  |-  ( z  =  suc  u  -> 
( F `  z
)  =  ( F `
 suc  u )
)
65eleq1d 2502 . . 3  |-  ( z  =  suc  u  -> 
( ( F `  z )  e.  A  <->  ( F `  suc  u
)  e.  A ) )
7 omsson 4849 . . . . . 6  |-  om  C_  On
8 sstr 3356 . . . . . 6  |-  ( ( A  C_  om  /\  om  C_  On )  ->  A  C_  On )
97, 8mpan2 653 . . . . 5  |-  ( A 
C_  om  ->  A  C_  On )
10 peano1 4864 . . . . . . . . 9  |-  (/)  e.  om
11 eleq1 2496 . . . . . . . . . . 11  |-  ( w  =  (/)  ->  ( w  e.  v  <->  (/)  e.  v ) )
1211rexbidv 2726 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( E. v  e.  A  w  e.  v  <->  E. v  e.  A  (/)  e.  v ) )
1312rspcv 3048 . . . . . . . . 9  |-  ( (/)  e.  om  ->  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  E. v  e.  A  (/) 
e.  v ) )
1410, 13ax-mp 8 . . . . . . . 8  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  E. v  e.  A  (/) 
e.  v )
15 df-rex 2711 . . . . . . . 8  |-  ( E. v  e.  A  (/)  e.  v  <->  E. v ( v  e.  A  /\  (/)  e.  v ) )
1614, 15sylib 189 . . . . . . 7  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  E. v ( v  e.  A  /\  (/)  e.  v ) )
17 exsimpl 1602 . . . . . . 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 3637 . . . . . 6  |-  ( A  =/=  (/)  <->  E. v  v  e.  A )
2018, 19sylibr 204 . . . . 5  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  A  =/=  (/) )
21 onint 4775 . . . . 5  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
229, 20, 21syl2an 464 . . . 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 5729 . . . . . . 7  |-  ( F `
 (/) )  =  ( ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om ) `  (/) )
25 fr0g 6693 . . . . . . 7  |-  ( |^| A  e.  A  ->  ( ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om ) `  (/) )  = 
|^| A )
2624, 25syl5req 2481 . . . . . 6  |-  ( |^| A  e.  A  ->  |^| A  =  ( F `
 (/) ) )
2726eleq1d 2502 . . . . 5  |-  ( |^| A  e.  A  ->  (
|^| A  e.  A  <->  ( F `  (/) )  e.  A ) )
2827ibi 233 . . . 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 7359 . . . . 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 4646 . . . . . . . . . . . 12  |-  ( y  =  x  ->  suc  y  =  suc  x )
3231difeq2d 3465 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( A  \  suc  y )  =  ( A  \  suc  x ) )
3332inteqd 4055 . . . . . . . . . 10  |-  ( y  =  x  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  x )
)
34 suceq 4646 . . . . . . . . . . . 12  |-  ( y  =  ( F `  u )  ->  suc  y  =  suc  ( F `
 u ) )
3534difeq2d 3465 . . . . . . . . . . 11  |-  ( y  =  ( F `  u )  ->  ( A  \  suc  y )  =  ( A  \  suc  ( F `  u
) ) )
3635inteqd 4055 . . . . . . . . . 10  |-  ( y  =  ( F `  u )  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  ( F `  u ) ) )
3723, 33, 36frsucmpt2 6697 . . . . . . . . 9  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  ( F `  suc  u )  =  |^| ( A  \  suc  ( F `  u )
) )
3837eqcomd 2441 . . . . . . . 8  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  |^| ( A  \  suc  ( F `  u
) )  =  ( F `  suc  u
) )
3938eleq1d 2502 . . . . . . 7  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  ( |^| ( A  \  suc  ( F `
 u ) )  e.  A  <->  ( F `  suc  u )  e.  A ) )
4039ex 424 . . . . . 6  |-  ( u  e.  om  ->  ( |^| ( A  \  suc  ( F `  u ) )  e.  A  -> 
( |^| ( A  \  suc  ( F `  u
) )  e.  A  <->  ( F `  suc  u
)  e.  A ) ) )
4140ibd 235 . . . . 5  |-  ( u  e.  om  ->  ( |^| ( A  \  suc  ( F `  u ) )  e.  A  -> 
( F `  suc  u )  e.  A
) )
4230, 41syl5 30 . . . 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
) )
4342exp3a 426 . . 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 4873 . 2  |-  ( z  e.  om  ->  (
( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  ( F `  z )  e.  A ) )
4544com12 29 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   _Vcvv 2956    \ cdif 3317    C_ wss 3320   (/)c0 3628   |^|cint 4050    e. cmpt 4266   Oncon0 4581   suc csuc 4583   omcom 4845    |` cres 4880   ` cfv 5454   reccrdg 6667
This theorem is referenced by:  unblem3  7361  unblem4  7362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668
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