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

Theorem unblem2 7829
Description: Lemma for unbnn 7832. 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 5870 . . . 4  |-  ( z  =  (/)  ->  ( F `
 z )  =  ( F `  (/) ) )
21eleq1d 2515 . . 3  |-  ( z  =  (/)  ->  ( ( F `  z )  e.  A  <->  ( F `  (/) )  e.  A
) )
3 fveq2 5870 . . . 4  |-  ( z  =  u  ->  ( F `  z )  =  ( F `  u ) )
43eleq1d 2515 . . 3  |-  ( z  =  u  ->  (
( F `  z
)  e.  A  <->  ( F `  u )  e.  A
) )
5 fveq2 5870 . . . 4  |-  ( z  =  suc  u  -> 
( F `  z
)  =  ( F `
 suc  u )
)
65eleq1d 2515 . . 3  |-  ( z  =  suc  u  -> 
( ( F `  z )  e.  A  <->  ( F `  suc  u
)  e.  A ) )
7 omsson 6701 . . . . . 6  |-  om  C_  On
8 sstr 3442 . . . . . 6  |-  ( ( A  C_  om  /\  om  C_  On )  ->  A  C_  On )
97, 8mpan2 678 . . . . 5  |-  ( A 
C_  om  ->  A  C_  On )
10 peano1 6717 . . . . . . . . 9  |-  (/)  e.  om
11 eleq1 2519 . . . . . . . . . . 11  |-  ( w  =  (/)  ->  ( w  e.  v  <->  (/)  e.  v ) )
1211rexbidv 2903 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( E. v  e.  A  w  e.  v  <->  E. v  e.  A  (/)  e.  v ) )
1312rspcv 3148 . . . . . . . . 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 2745 . . . . . . . 8  |-  ( E. v  e.  A  (/)  e.  v  <->  E. v ( v  e.  A  /\  (/)  e.  v ) )
1614, 15sylib 200 . . . . . . 7  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  E. v ( v  e.  A  /\  (/)  e.  v ) )
17 exsimpl 1731 . . . . . . 7  |-  ( E. v ( v  e.  A  /\  (/)  e.  v )  ->  E. v 
v  e.  A )
1816, 17syl 17 . . . . . 6  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  E. v  v  e.  A )
19 n0 3743 . . . . . 6  |-  ( A  =/=  (/)  <->  E. v  v  e.  A )
2018, 19sylibr 216 . . . . 5  |-  ( A. w  e.  om  E. v  e.  A  w  e.  v  ->  A  =/=  (/) )
21 onint 6627 . . . . 5  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
229, 20, 21syl2an 480 . . . 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 5871 . . . . . . 7  |-  ( F `
 (/) )  =  ( ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om ) `  (/) )
25 fr0g 7158 . . . . . . 7  |-  ( |^| A  e.  A  ->  ( ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om ) `  (/) )  = 
|^| A )
2624, 25syl5req 2500 . . . . . 6  |-  ( |^| A  e.  A  ->  |^| A  =  ( F `
 (/) ) )
2726eleq1d 2515 . . . . 5  |-  ( |^| A  e.  A  ->  (
|^| A  e.  A  <->  ( F `  (/) )  e.  A ) )
2827ibi 245 . . . 4  |-  ( |^| A  e.  A  ->  ( F `  (/) )  e.  A )
2922, 28syl 17 . . 3  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  ( F `  (/) )  e.  A )
30 unblem1 7828 . . . . 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 5491 . . . . . . . . . . . 12  |-  ( y  =  x  ->  suc  y  =  suc  x )
3231difeq2d 3553 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( A  \  suc  y )  =  ( A  \  suc  x ) )
3332inteqd 4242 . . . . . . . . . 10  |-  ( y  =  x  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  x )
)
34 suceq 5491 . . . . . . . . . . . 12  |-  ( y  =  ( F `  u )  ->  suc  y  =  suc  ( F `
 u ) )
3534difeq2d 3553 . . . . . . . . . . 11  |-  ( y  =  ( F `  u )  ->  ( A  \  suc  y )  =  ( A  \  suc  ( F `  u
) ) )
3635inteqd 4242 . . . . . . . . . 10  |-  ( y  =  ( F `  u )  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  ( F `  u ) ) )
3723, 33, 36frsucmpt2 7162 . . . . . . . . 9  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  ( F `  suc  u )  =  |^| ( A  \  suc  ( F `  u )
) )
3837eqcomd 2459 . . . . . . . 8  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  |^| ( A  \  suc  ( F `  u
) )  =  ( F `  suc  u
) )
3938eleq1d 2515 . . . . . . 7  |-  ( ( u  e.  om  /\  |^| ( A  \  suc  ( F `  u ) )  e.  A )  ->  ( |^| ( A  \  suc  ( F `
 u ) )  e.  A  <->  ( F `  suc  u )  e.  A ) )
4039ex 436 . . . . . 6  |-  ( u  e.  om  ->  ( |^| ( A  \  suc  ( F `  u ) )  e.  A  -> 
( |^| ( A  \  suc  ( F `  u
) )  e.  A  <->  ( F `  suc  u
)  e.  A ) ) )
4140ibd 247 . . . . 5  |-  ( u  e.  om  ->  ( |^| ( A  \  suc  ( F `  u ) )  e.  A  -> 
( F `  suc  u )  e.  A
) )
4230, 41syl5 33 . . . 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 438 . . 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 6726 . 2  |-  ( z  e.  om  ->  (
( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  ( F `  z )  e.  A ) )
4544com12 32 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 188    /\ wa 371    = wceq 1446   E.wex 1665    e. wcel 1889    =/= wne 2624   A.wral 2739   E.wrex 2740   _Vcvv 3047    \ cdif 3403    C_ wss 3406   (/)c0 3733   |^|cint 4237    |-> cmpt 4464    |` cres 4839   Oncon0 5426   suc csuc 5428   ` cfv 5585   omcom 6697   reccrdg 7132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133
This theorem is referenced by:  unblem3  7830  unblem4  7831
  Copyright terms: Public domain W3C validator