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Theorem unblem3 7566
Description: Lemma for unbnn 7568. The value of the function  F is less than its value at a successor. (Contributed by NM, 3-Dec-2003.)
Hypothesis
Ref Expression
unblem.2  |-  F  =  ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om )
Assertion
Ref Expression
unblem3  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
z  e.  om  ->  ( F `  z )  e.  ( F `  suc  z ) ) )
Distinct variable groups:    w, v, x, z, A    v, F, w, z
Allowed substitution hint:    F( x)

Proof of Theorem unblem3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 unblem.2 . . . . . . 7  |-  F  =  ( rec ( ( x  e.  _V  |->  |^| ( A  \  suc  x ) ) , 
|^| A )  |`  om )
21unblem2 7565 . . . . . 6  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
z  e.  om  ->  ( F `  z )  e.  A ) )
32imp 429 . . . . 5  |-  ( ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  /\  z  e.  om )  ->  ( F `  z )  e.  A )
4 omsson 6480 . . . . . . . 8  |-  om  C_  On
5 sstr 3364 . . . . . . . 8  |-  ( ( A  C_  om  /\  om  C_  On )  ->  A  C_  On )
64, 5mpan2 671 . . . . . . 7  |-  ( A 
C_  om  ->  A  C_  On )
7 ssel 3350 . . . . . . . 8  |-  ( A 
C_  On  ->  ( ( F `  z )  e.  A  ->  ( F `  z )  e.  On ) )
87anc2li 557 . . . . . . 7  |-  ( A 
C_  On  ->  ( ( F `  z )  e.  A  ->  ( A  C_  On  /\  ( F `  z )  e.  On ) ) )
96, 8syl 16 . . . . . 6  |-  ( A 
C_  om  ->  ( ( F `  z )  e.  A  ->  ( A  C_  On  /\  ( F `  z )  e.  On ) ) )
109ad2antrr 725 . . . . 5  |-  ( ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  /\  z  e.  om )  ->  (
( F `  z
)  e.  A  -> 
( A  C_  On  /\  ( F `  z
)  e.  On ) ) )
113, 10mpd 15 . . . 4  |-  ( ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  /\  z  e.  om )  ->  ( A  C_  On  /\  ( F `  z )  e.  On ) )
12 onmindif 4808 . . . 4  |-  ( ( A  C_  On  /\  ( F `  z )  e.  On )  ->  ( F `  z )  e.  |^| ( A  \  suc  ( F `  z
) ) )
1311, 12syl 16 . . 3  |-  ( ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  /\  z  e.  om )  ->  ( F `  z )  e.  |^| ( A  \  suc  ( F `  z
) ) )
14 unblem1 7564 . . . . . . 7  |-  ( ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  /\  ( F `  z )  e.  A )  ->  |^| ( A  \  suc  ( F `
 z ) )  e.  A )
1514ex 434 . . . . . 6  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
( F `  z
)  e.  A  ->  |^| ( A  \  suc  ( F `  z ) )  e.  A ) )
162, 15syld 44 . . . . 5  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
z  e.  om  ->  |^| ( A  \  suc  ( F `  z ) )  e.  A ) )
17 suceq 4784 . . . . . . . . 9  |-  ( y  =  x  ->  suc  y  =  suc  x )
1817difeq2d 3474 . . . . . . . 8  |-  ( y  =  x  ->  ( A  \  suc  y )  =  ( A  \  suc  x ) )
1918inteqd 4133 . . . . . . 7  |-  ( y  =  x  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  x )
)
20 suceq 4784 . . . . . . . . 9  |-  ( y  =  ( F `  z )  ->  suc  y  =  suc  ( F `
 z ) )
2120difeq2d 3474 . . . . . . . 8  |-  ( y  =  ( F `  z )  ->  ( A  \  suc  y )  =  ( A  \  suc  ( F `  z
) ) )
2221inteqd 4133 . . . . . . 7  |-  ( y  =  ( F `  z )  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  ( F `  z ) ) )
231, 19, 22frsucmpt2 6895 . . . . . 6  |-  ( ( z  e.  om  /\  |^| ( A  \  suc  ( F `  z ) )  e.  A )  ->  ( F `  suc  z )  =  |^| ( A  \  suc  ( F `  z )
) )
2423ex 434 . . . . 5  |-  ( z  e.  om  ->  ( |^| ( A  \  suc  ( F `  z ) )  e.  A  -> 
( F `  suc  z )  =  |^| ( A  \  suc  ( F `  z )
) ) )
2516, 24sylcom 29 . . . 4  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
z  e.  om  ->  ( F `  suc  z
)  =  |^| ( A  \  suc  ( F `
 z ) ) ) )
2625imp 429 . . 3  |-  ( ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  /\  z  e.  om )  ->  ( F `  suc  z )  =  |^| ( A 
\  suc  ( F `  z ) ) )
2713, 26eleqtrrd 2520 . 2  |-  ( ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  /\  z  e.  om )  ->  ( F `  z )  e.  ( F `  suc  z ) )
2827ex 434 1  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
z  e.  om  ->  ( F `  z )  e.  ( F `  suc  z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   _Vcvv 2972    \ cdif 3325    C_ wss 3328   |^|cint 4128    e. cmpt 4350   Oncon0 4719   suc csuc 4721    |` cres 4842   ` cfv 5418   omcom 6476   reccrdg 6865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-om 6477  df-recs 6832  df-rdg 6866
This theorem is referenced by:  unblem4  7567
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