MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unblem2 Unicode version

Theorem unblem2 7155
Description: Lemma for unbnn 7158. 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 5563 . . . 4  |-  ( z  =  (/)  ->  ( F `
 z )  =  ( F `  (/) ) )
21eleq1d 2382 . . 3  |-  ( z  =  (/)  ->  ( ( F `  z )  e.  A  <->  ( F `  (/) )  e.  A
) )
3 fveq2 5563 . . . 4  |-  ( z  =  u  ->  ( F `  z )  =  ( F `  u ) )
43eleq1d 2382 . . 3  |-  ( z  =  u  ->  (
( F `  z
)  e.  A  <->  ( F `  u )  e.  A
) )
5 fveq2 5563 . . . 4  |-  ( z  =  suc  u  -> 
( F `  z
)  =  ( F `
 suc  u )
)
65eleq1d 2382 . . 3  |-  ( z  =  suc  u  -> 
( ( F `  z )  e.  A  <->  ( F `  suc  u
)  e.  A ) )
7 omsson 4697 . . . . . 6  |-  om  C_  On
8 sstr 3221 . . . . . 6  |-  ( ( A  C_  om  /\  om  C_  On )  ->  A  C_  On )
97, 8mpan2 652 . . . . 5  |-  ( A 
C_  om  ->  A  C_  On )
10 peano1 4712 . . . . . . . . 9  |-  (/)  e.  om
11 eleq1 2376 . . . . . . . . . . 11  |-  ( w  =  (/)  ->  ( w  e.  v  <->  (/)  e.  v ) )
1211rexbidv 2598 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( E. v  e.  A  w  e.  v  <->  E. v  e.  A  (/)  e.  v ) )
1312rspcv 2914 . . . . . . . . 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 2583 . . . . . . . 8  |-  ( E. v  e.  A  (/)  e.  v  <->  E. v ( v  e.  A  /\  (/)  e.  v ) )
1614, 15sylib 188 . . . . . . 7  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  E. v ( v  e.  A  /\  (/)  e.  v ) )
17 exsimpl 1583 . . . . . . 7  |-  ( E. v ( v  e.  A  /\  (/)  e.  v )  ->  E. v 
v  e.  A )
1816, 17syl 15 . . . . . 6  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  E. v  v  e.  A )
19 n0 3498 . . . . . 6  |-  ( A  =/=  (/)  <->  E. v  v  e.  A )
2018, 19sylibr 203 . . . . 5  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  A  =/=  (/) )
21 onint 4623 . . . . 5  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
229, 20, 21syl2an 463 . . . 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 5564 . . . . . . 7  |-  ( F `
 (/) )  =  ( ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om ) `  (/) )
25 fr0g 6490 . . . . . . 7  |-  ( |^| A  e.  A  ->  ( ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om ) `  (/) )  = 
|^| A )
2624, 25syl5req 2361 . . . . . 6  |-  ( |^| A  e.  A  ->  |^| A  =  ( F `
 (/) ) )
2726eleq1d 2382 . . . . 5  |-  ( |^| A  e.  A  ->  (
|^| A  e.  A  <->  ( F `  (/) )  e.  A ) )
2827ibi 232 . . . 4  |-  ( |^| A  e.  A  ->  ( F `  (/) )  e.  A )
2922, 28syl 15 . . 3  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  ( F `  (/) )  e.  A )
30 unblem1 7154 . . . . 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 4494 . . . . . . . . . . . 12  |-  ( y  =  x  ->  suc  y  =  suc  x )
3231difeq2d 3328 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( A  \  suc  y )  =  ( A  \  suc  x ) )
3332inteqd 3904 . . . . . . . . . 10  |-  ( y  =  x  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  x )
)
34 suceq 4494 . . . . . . . . . . . 12  |-  ( y  =  ( F `  u )  ->  suc  y  =  suc  ( F `
 u ) )
3534difeq2d 3328 . . . . . . . . . . 11  |-  ( y  =  ( F `  u )  ->  ( A  \  suc  y )  =  ( A  \  suc  ( F `  u
) ) )
3635inteqd 3904 . . . . . . . . . 10  |-  ( y  =  ( F `  u )  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  ( F `  u ) ) )
3723, 33, 36frsucmpt2 6494 . . . . . . . . 9  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  ( F `  suc  u )  =  |^| ( A  \  suc  ( F `  u )
) )
3837eqcomd 2321 . . . . . . . 8  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  |^| ( A  \  suc  ( F `  u
) )  =  ( F `  suc  u
) )
3938eleq1d 2382 . . . . . . 7  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  ( |^| ( A  \  suc  ( F `
 u ) )  e.  A  <->  ( F `  suc  u )  e.  A ) )
4039ex 423 . . . . . 6  |-  ( u  e.  om  ->  ( |^| ( A  \  suc  ( F `  u ) )  e.  A  -> 
( |^| ( A  \  suc  ( F `  u
) )  e.  A  <->  ( F `  suc  u
)  e.  A ) ) )
4140ibd 234 . . . . 5  |-  ( u  e.  om  ->  ( |^| ( A  \  suc  ( F `  u ) )  e.  A  -> 
( F `  suc  u )  e.  A
) )
4230, 41syl5 28 . . . 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 425 . . 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 4721 . 2  |-  ( z  e.  om  ->  (
( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  ( F `  z )  e.  A ) )
4544com12 27 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 176    /\ wa 358   E.wex 1532    = wceq 1633    e. wcel 1701    =/= wne 2479   A.wral 2577   E.wrex 2578   _Vcvv 2822    \ cdif 3183    C_ wss 3186   (/)c0 3489   |^|cint 3899    e. cmpt 4114   Oncon0 4429   suc csuc 4431   omcom 4693    |` cres 4728   ` cfv 5292   reccrdg 6464
This theorem is referenced by:  unblem3  7156  unblem4  7157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-recs 6430  df-rdg 6465
  Copyright terms: Public domain W3C validator