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Theorem unblem3 7258
Description: Lemma for unbnn 7260. 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 7257 . . . . . 6  |-  ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
z  e.  om  ->  ( F `  z )  e.  A ) )
32imp 418 . . . . 5  |-  ( ( ( A  C_  om  /\  A. w  e.  om  E. v  e.  A  w  e.  v )  /\  z  e.  om )  ->  ( F `  z )  e.  A )
4 omsson 4763 . . . . . . . 8  |-  om  C_  On
5 sstr 3273 . . . . . . . 8  |-  ( ( A  C_  om  /\  om  C_  On )  ->  A  C_  On )
64, 5mpan2 652 . . . . . . 7  |-  ( A 
C_  om  ->  A  C_  On )
7 ssel 3260 . . . . . . . 8  |-  ( A 
C_  On  ->  ( ( F `  z )  e.  A  ->  ( F `  z )  e.  On ) )
87anc2li 540 . . . . . . 7  |-  ( A 
C_  On  ->  ( ( F `  z )  e.  A  ->  ( A  C_  On  /\  ( F `  z )  e.  On ) ) )
96, 8syl 15 . . . . . 6  |-  ( A 
C_  om  ->  ( ( F `  z )  e.  A  ->  ( A  C_  On  /\  ( F `  z )  e.  On ) ) )
109ad2antrr 706 . . . . 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 14 . . . 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 4585 . . . 4  |-  ( ( A  C_  On  /\  ( F `  z )  e.  On )  ->  ( F `  z )  e.  |^| ( A  \  suc  ( F `  z
) ) )
1311, 12syl 15 . . 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 7256 . . . . . . 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 423 . . . . . 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 40 . . . . 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 4560 . . . . . . . . 9  |-  ( y  =  x  ->  suc  y  =  suc  x )
1817difeq2d 3381 . . . . . . . 8  |-  ( y  =  x  ->  ( A  \  suc  y )  =  ( A  \  suc  x ) )
1918inteqd 3969 . . . . . . 7  |-  ( y  =  x  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  x )
)
20 suceq 4560 . . . . . . . . 9  |-  ( y  =  ( F `  z )  ->  suc  y  =  suc  ( F `
 z ) )
2120difeq2d 3381 . . . . . . . 8  |-  ( y  =  ( F `  z )  ->  ( A  \  suc  y )  =  ( A  \  suc  ( F `  z
) ) )
2221inteqd 3969 . . . . . . 7  |-  ( y  =  ( F `  z )  ->  |^| ( A  \  suc  y )  =  |^| ( A 
\  suc  ( F `  z ) ) )
231, 19, 22frsucmpt2 6594 . . . . . 6  |-  ( ( z  e.  om  /\  |^| ( A  \  suc  ( F `  z ) )  e.  A )  ->  ( F `  suc  z )  =  |^| ( A  \  suc  ( F `  z )
) )
2423ex 423 . . . . 5  |-  ( z  e.  om  ->  ( |^| ( A  \  suc  ( F `  z ) )  e.  A  -> 
( F `  suc  z )  =  |^| ( A  \  suc  ( F `  z )
) ) )
2516, 24sylcom 25 . . . 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 418 . . 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 2443 . 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 423 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 358    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629   _Vcvv 2873    \ cdif 3235    C_ wss 3238   |^|cint 3964    e. cmpt 4179   Oncon0 4495   suc csuc 4497   omcom 4759    |` cres 4794   ` cfv 5358   reccrdg 6564
This theorem is referenced by:  unblem4  7259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-recs 6530  df-rdg 6565
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