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Theorem unblem3 7562
Description: Lemma for unbnn 7564. 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 7561 . . . . . 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 6479 . . . . . . . 8  |-  om  C_  On
5 sstr 3361 . . . . . . . 8  |-  ( ( A  C_  om  /\  om  C_  On )  ->  A  C_  On )
64, 5mpan2 666 . . . . . . 7  |-  ( A 
C_  om  ->  A  C_  On )
7 ssel 3347 . . . . . . . 8  |-  ( A 
C_  On  ->  ( ( F `  z )  e.  A  ->  ( F `  z )  e.  On ) )
87anc2li 554 . . . . . . 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 720 . . . . 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 4804 . . . 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 7560 . . . . . . 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 4780 . . . . . . . . 9  |-  ( y  =  x  ->  suc  y  =  suc  x )
1817difeq2d 3471 . . . . . . . 8  |-  ( y  =  x  ->  ( A  \  suc  y )  =  ( A  \  suc  x ) )
1918inteqd 4130 . . . . . . 7  |-  ( y  =  x  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  x )
)
20 suceq 4780 . . . . . . . . 9  |-  ( y  =  ( F `  z )  ->  suc  y  =  suc  ( F `
 z ) )
2120difeq2d 3471 . . . . . . . 8  |-  ( y  =  ( F `  z )  ->  ( A  \  suc  y )  =  ( A  \  suc  ( F `  z
) ) )
2221inteqd 4130 . . . . . . 7  |-  ( y  =  ( F `  z )  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  ( F `  z ) ) )
231, 19, 22frsucmpt2 6891 . . . . . 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 2518 . 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 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   _Vcvv 2970    \ cdif 3322    C_ wss 3325   |^|cint 4125    e. cmpt 4347   Oncon0 4715   suc csuc 4717    |` cres 4838   ` cfv 5415   omcom 6475   reccrdg 6861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-recs 6828  df-rdg 6862
This theorem is referenced by:  unblem4  7563
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