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Theorem unblem3 7766
Description: Lemma for unbnn 7768. 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 7765 . . . . . 6  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
z  e.  om  ->  ( F `  z )  e.  A ) )
32imp 427 . . . . 5  |-  ( ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  /\  z  e.  om )  ->  ( F `  z )  e.  A )
4 omsson 6677 . . . . . . . 8  |-  om  C_  On
5 sstr 3497 . . . . . . . 8  |-  ( ( A  C_  om  /\  om  C_  On )  ->  A  C_  On )
64, 5mpan2 669 . . . . . . 7  |-  ( A 
C_  om  ->  A  C_  On )
7 ssel 3483 . . . . . . . 8  |-  ( A 
C_  On  ->  ( ( F `  z )  e.  A  ->  ( F `  z )  e.  On ) )
87anc2li 555 . . . . . . 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 723 . . . . 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 4956 . . . 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 7764 . . . . . . 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 432 . . . . . 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 4932 . . . . . . . . 9  |-  ( y  =  x  ->  suc  y  =  suc  x )
1817difeq2d 3608 . . . . . . . 8  |-  ( y  =  x  ->  ( A  \  suc  y )  =  ( A  \  suc  x ) )
1918inteqd 4276 . . . . . . 7  |-  ( y  =  x  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  x )
)
20 suceq 4932 . . . . . . . . 9  |-  ( y  =  ( F `  z )  ->  suc  y  =  suc  ( F `
 z ) )
2120difeq2d 3608 . . . . . . . 8  |-  ( y  =  ( F `  z )  ->  ( A  \  suc  y )  =  ( A  \  suc  ( F `  z
) ) )
2221inteqd 4276 . . . . . . 7  |-  ( y  =  ( F `  z )  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  ( F `  z ) ) )
231, 19, 22frsucmpt2 7097 . . . . . 6  |-  ( ( z  e.  om  /\  |^| ( A  \  suc  ( F `  z ) )  e.  A )  ->  ( F `  suc  z )  =  |^| ( A  \  suc  ( F `  z )
) )
2423ex 432 . . . . 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 427 . . 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 2545 . 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 432 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    \ cdif 3458    C_ wss 3461   |^|cint 4271    |-> cmpt 4497   Oncon0 4867   suc csuc 4869    |` cres 4990   ` cfv 5570   omcom 6673   reccrdg 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068
This theorem is referenced by:  unblem4  7767
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