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Theorem unblem3 7364
Description: Lemma for unbnn 7366. 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 7363 . . . . . 6  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
z  e.  om  ->  ( F `  z )  e.  A ) )
32imp 420 . . . . 5  |-  ( ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  /\  z  e.  om )  ->  ( F `  z )  e.  A )
4 omsson 4852 . . . . . . . 8  |-  om  C_  On
5 sstr 3358 . . . . . . . 8  |-  ( ( A  C_  om  /\  om  C_  On )  ->  A  C_  On )
64, 5mpan2 654 . . . . . . 7  |-  ( A 
C_  om  ->  A  C_  On )
7 ssel 3344 . . . . . . . 8  |-  ( A 
C_  On  ->  ( ( F `  z )  e.  A  ->  ( F `  z )  e.  On ) )
87anc2li 542 . . . . . . 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 708 . . . . 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 4674 . . . 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 7362 . . . . . . 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 425 . . . . . 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 43 . . . . 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 4649 . . . . . . . . 9  |-  ( y  =  x  ->  suc  y  =  suc  x )
1817difeq2d 3467 . . . . . . . 8  |-  ( y  =  x  ->  ( A  \  suc  y )  =  ( A  \  suc  x ) )
1918inteqd 4057 . . . . . . 7  |-  ( y  =  x  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  x )
)
20 suceq 4649 . . . . . . . . 9  |-  ( y  =  ( F `  z )  ->  suc  y  =  suc  ( F `
 z ) )
2120difeq2d 3467 . . . . . . . 8  |-  ( y  =  ( F `  z )  ->  ( A  \  suc  y )  =  ( A  \  suc  ( F `  z
) ) )
2221inteqd 4057 . . . . . . 7  |-  ( y  =  ( F `  z )  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  ( F `  z ) ) )
231, 19, 22frsucmpt2 6700 . . . . . 6  |-  ( ( z  e.  om  /\  |^| ( A  \  suc  ( F `  z ) )  e.  A )  ->  ( F `  suc  z )  =  |^| ( A  \  suc  ( F `  z )
) )
2423ex 425 . . . . 5  |-  ( z  e.  om  ->  ( |^| ( A  \  suc  ( F `  z ) )  e.  A  -> 
( F `  suc  z )  =  |^| ( A  \  suc  ( F `  z )
) ) )
2516, 24sylcom 28 . . . 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 420 . . 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 2515 . 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 425 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    \ cdif 3319    C_ wss 3322   |^|cint 4052    e. cmpt 4269   Oncon0 4584   suc csuc 4586   omcom 4848    |` cres 4883   ` cfv 5457   reccrdg 6670
This theorem is referenced by:  unblem4  7365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-recs 6636  df-rdg 6671
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