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Theorem unblem2 7688
Description: Lemma for unbnn 7691. 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 5774 . . . 4  |-  ( z  =  (/)  ->  ( F `
 z )  =  ( F `  (/) ) )
21eleq1d 2451 . . 3  |-  ( z  =  (/)  ->  ( ( F `  z )  e.  A  <->  ( F `  (/) )  e.  A
) )
3 fveq2 5774 . . . 4  |-  ( z  =  u  ->  ( F `  z )  =  ( F `  u ) )
43eleq1d 2451 . . 3  |-  ( z  =  u  ->  (
( F `  z
)  e.  A  <->  ( F `  u )  e.  A
) )
5 fveq2 5774 . . . 4  |-  ( z  =  suc  u  -> 
( F `  z
)  =  ( F `
 suc  u )
)
65eleq1d 2451 . . 3  |-  ( z  =  suc  u  -> 
( ( F `  z )  e.  A  <->  ( F `  suc  u
)  e.  A ) )
7 omsson 6603 . . . . . 6  |-  om  C_  On
8 sstr 3425 . . . . . 6  |-  ( ( A  C_  om  /\  om  C_  On )  ->  A  C_  On )
97, 8mpan2 669 . . . . 5  |-  ( A 
C_  om  ->  A  C_  On )
10 peano1 6618 . . . . . . . . 9  |-  (/)  e.  om
11 eleq1 2454 . . . . . . . . . . 11  |-  ( w  =  (/)  ->  ( w  e.  v  <->  (/)  e.  v ) )
1211rexbidv 2893 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( E. v  e.  A  w  e.  v  <->  E. v  e.  A  (/)  e.  v ) )
1312rspcv 3131 . . . . . . . . 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 2738 . . . . . . . 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 1685 . . . . . . 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 3721 . . . . . 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 6529 . . . . 5  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
229, 20, 21syl2an 475 . . . 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 5775 . . . . . . 7  |-  ( F `
 (/) )  =  ( ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om ) `  (/) )
25 fr0g 7019 . . . . . . 7  |-  ( |^| A  e.  A  ->  ( ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om ) `  (/) )  = 
|^| A )
2624, 25syl5req 2436 . . . . . 6  |-  ( |^| A  e.  A  ->  |^| A  =  ( F `
 (/) ) )
2726eleq1d 2451 . . . . 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 7687 . . . . 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 4857 . . . . . . . . . . . 12  |-  ( y  =  x  ->  suc  y  =  suc  x )
3231difeq2d 3536 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( A  \  suc  y )  =  ( A  \  suc  x ) )
3332inteqd 4204 . . . . . . . . . 10  |-  ( y  =  x  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  x )
)
34 suceq 4857 . . . . . . . . . . . 12  |-  ( y  =  ( F `  u )  ->  suc  y  =  suc  ( F `
 u ) )
3534difeq2d 3536 . . . . . . . . . . 11  |-  ( y  =  ( F `  u )  ->  ( A  \  suc  y )  =  ( A  \  suc  ( F `  u
) ) )
3635inteqd 4204 . . . . . . . . . 10  |-  ( y  =  ( F `  u )  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  ( F `  u ) ) )
3723, 33, 36frsucmpt2 7023 . . . . . . . . 9  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  ( F `  suc  u )  =  |^| ( A  \  suc  ( F `  u )
) )
3837eqcomd 2390 . . . . . . . 8  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  |^| ( A  \  suc  ( F `  u
) )  =  ( F `  suc  u
) )
3938eleq1d 2451 . . . . . . 7  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  ( |^| ( A  \  suc  ( F `
 u ) )  e.  A  <->  ( F `  suc  u )  e.  A ) )
4039ex 432 . . . . . 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 434 . . 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 6627 . 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 367    = wceq 1399   E.wex 1620    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733   _Vcvv 3034    \ cdif 3386    C_ wss 3389   (/)c0 3711   |^|cint 4199    |-> cmpt 4425   Oncon0 4792   suc csuc 4794    |` cres 4915   ` cfv 5496   omcom 6599   reccrdg 6993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-om 6600  df-recs 6960  df-rdg 6994
This theorem is referenced by:  unblem3  7689  unblem4  7690
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