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Theorem ramub2 13269
Description: It is sufficient to check the Ramsey property on finite sets of size equal to the upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
rami.c  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
rami.m  |-  ( ph  ->  M  e.  NN0 )
rami.r  |-  ( ph  ->  R  e.  V )
rami.f  |-  ( ph  ->  F : R --> NN0 )
ramub2.n  |-  ( ph  ->  N  e.  NN0 )
ramub2.i  |-  ( (
ph  /\  ( ( # `
 s )  =  N  /\  f : ( s C M ) --> R ) )  ->  E. c  e.  R  E. x  e.  ~P  s ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  ( `' f " {
c } ) ) )
Assertion
Ref Expression
ramub2  |-  ( ph  ->  ( M Ramsey  F )  <_  N )
Distinct variable groups:    f, c,
s, x, C    ph, c,
f, s, x    F, c, f, s, x    a,
b, c, f, i, s, x, M    R, c, f, s, x    N, a, c, f, i, s, x    V, c, f, s, x
Allowed substitution hints:    ph( i, a, b)    C( i, a, b)    R( i, a, b)    F( i, a, b)    N( b)    V( i, a, b)

Proof of Theorem ramub2
Dummy variables  g 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rami.c . 2  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
2 rami.m . 2  |-  ( ph  ->  M  e.  NN0 )
3 rami.r . 2  |-  ( ph  ->  R  e.  V )
4 rami.f . 2  |-  ( ph  ->  F : R --> NN0 )
5 ramub2.n . 2  |-  ( ph  ->  N  e.  NN0 )
65adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  N  e.  NN0 )
7 hashfz1 11517 . . . . . . 7  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
86, 7syl 15 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( # `  (
1 ... N ) )  =  N )
9 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  N  <_  ( # `
 t ) )
108, 9eqbrtrd 4145 . . . . 5  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( # `  (
1 ... N ) )  <_  ( # `  t
) )
11 fzfid 11199 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( 1 ... N )  e.  Fin )
12 vex 2876 . . . . . 6  |-  t  e. 
_V
13 hashdom 11540 . . . . . 6  |-  ( ( ( 1 ... N
)  e.  Fin  /\  t  e.  _V )  ->  ( ( # `  (
1 ... N ) )  <_  ( # `  t
)  <->  ( 1 ... N )  ~<_  t ) )
1411, 12, 13sylancl 643 . . . . 5  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( ( # `  ( 1 ... N
) )  <_  ( # `
 t )  <->  ( 1 ... N )  ~<_  t ) )
1510, 14mpbid 201 . . . 4  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( 1 ... N )  ~<_  t )
1612domen 7018 . . . 4  |-  ( ( 1 ... N )  ~<_  t  <->  E. s ( ( 1 ... N ) 
~~  s  /\  s  C_  t ) )
1715, 16sylib 188 . . 3  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  E. s ( ( 1 ... N ) 
~~  s  /\  s  C_  t ) )
18 simpll 730 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ph )
19 ensym 7053 . . . . . . . . . 10  |-  ( ( 1 ... N ) 
~~  s  ->  s  ~~  ( 1 ... N
) )
2019ad2antrl 708 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  s  ~~  ( 1 ... N
) )
21 hasheni 11519 . . . . . . . . 9  |-  ( s 
~~  ( 1 ... N )  ->  ( # `
 s )  =  ( # `  (
1 ... N ) ) )
2220, 21syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( # `
 s )  =  ( # `  (
1 ... N ) ) )
235ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  N  e.  NN0 )
2423, 7syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( # `
 ( 1 ... N ) )  =  N )
2522, 24eqtrd 2398 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( # `
 s )  =  N )
26 simplrr 737 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  g : ( t C M ) --> R )
2712a1i 10 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  t  e.  _V )
28 simprr 733 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  s  C_  t )
292ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  M  e.  NN0 )
301hashbcss 13259 . . . . . . . . 9  |-  ( ( t  e.  _V  /\  s  C_  t  /\  M  e.  NN0 )  ->  (
s C M ) 
C_  ( t C M ) )
3127, 28, 29, 30syl3anc 1183 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
s C M ) 
C_  ( t C M ) )
32 fssres 5514 . . . . . . . 8  |-  ( ( g : ( t C M ) --> R  /\  ( s C M )  C_  (
t C M ) )  ->  ( g  |`  ( s C M ) ) : ( s C M ) --> R )
3326, 31, 32syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
g  |`  ( s C M ) ) : ( s C M ) --> R )
34 vex 2876 . . . . . . . . 9  |-  g  e. 
_V
3534resex 5098 . . . . . . . 8  |-  ( g  |`  ( s C M ) )  e.  _V
36 feq1 5480 . . . . . . . . . . 11  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
f : ( s C M ) --> R  <-> 
( g  |`  (
s C M ) ) : ( s C M ) --> R ) )
3736anbi2d 684 . . . . . . . . . 10  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
( ( # `  s
)  =  N  /\  f : ( s C M ) --> R )  <-> 
( ( # `  s
)  =  N  /\  ( g  |`  (
s C M ) ) : ( s C M ) --> R ) ) )
3837anbi2d 684 . . . . . . . . 9  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
( ph  /\  (
( # `  s )  =  N  /\  f : ( s C M ) --> R ) )  <->  ( ph  /\  ( ( # `  s
)  =  N  /\  ( g  |`  (
s C M ) ) : ( s C M ) --> R ) ) ) )
39 cnveq 4958 . . . . . . . . . . . . . 14  |-  ( f  =  ( g  |`  ( s C M ) )  ->  `' f  =  `' (
g  |`  ( s C M ) ) )
4039imaeq1d 5114 . . . . . . . . . . . . 13  |-  ( f  =  ( g  |`  ( s C M ) )  ->  ( `' f " {
c } )  =  ( `' ( g  |`  ( s C M ) ) " {
c } ) )
41 cnvresima 5265 . . . . . . . . . . . . 13  |-  ( `' ( g  |`  (
s C M ) ) " { c } )  =  ( ( `' g " { c } )  i^i  ( s C M ) )
4240, 41syl6eq 2414 . . . . . . . . . . . 12  |-  ( f  =  ( g  |`  ( s C M ) )  ->  ( `' f " {
c } )  =  ( ( `' g
" { c } )  i^i  ( s C M ) ) )
4342sseq2d 3292 . . . . . . . . . . 11  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
( x C M )  C_  ( `' f " { c } )  <->  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) ) )
4443anbi2d 684 . . . . . . . . . 10  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
( ( F `  c )  <_  ( # `
 x )  /\  ( x C M )  C_  ( `' f " { c } ) )  <->  ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) ) ) )
45442rexbidv 2671 . . . . . . . . 9  |-  ( f  =  ( g  |`  ( s C M ) )  ->  ( E. c  e.  R  E. x  e.  ~P  s ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  ( `' f " {
c } ) )  <->  E. c  e.  R  E. x  e.  ~P  s ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) ) ) )
4638, 45imbi12d 311 . . . . . . . 8  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
( ( ph  /\  ( ( # `  s
)  =  N  /\  f : ( s C M ) --> R ) )  ->  E. c  e.  R  E. x  e.  ~P  s ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' f " {
c } ) ) )  <->  ( ( ph  /\  ( ( # `  s
)  =  N  /\  ( g  |`  (
s C M ) ) : ( s C M ) --> R ) )  ->  E. c  e.  R  E. x  e.  ~P  s ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) ) ) ) )
47 ramub2.i . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 s )  =  N  /\  f : ( s C M ) --> R ) )  ->  E. c  e.  R  E. x  e.  ~P  s ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  ( `' f " {
c } ) ) )
4835, 46, 47vtocl 2923 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 s )  =  N  /\  ( g  |`  ( s C M ) ) : ( s C M ) --> R ) )  ->  E. c  e.  R  E. x  e.  ~P  s ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) ) )
4918, 25, 33, 48syl12anc 1181 . . . . . 6  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  E. c  e.  R  E. x  e.  ~P  s ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) ) )
50 sstr 3273 . . . . . . . . . . . 12  |-  ( ( x  C_  s  /\  s  C_  t )  ->  x  C_  t )
5150expcom 424 . . . . . . . . . . 11  |-  ( s 
C_  t  ->  (
x  C_  s  ->  x 
C_  t ) )
5251ad2antll 709 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
x  C_  s  ->  x 
C_  t ) )
53 vex 2876 . . . . . . . . . . 11  |-  x  e. 
_V
5453elpw 3720 . . . . . . . . . 10  |-  ( x  e.  ~P s  <->  x  C_  s
)
5553elpw 3720 . . . . . . . . . 10  |-  ( x  e.  ~P t  <->  x  C_  t
)
5652, 54, 553imtr4g 261 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
x  e.  ~P s  ->  x  e.  ~P t
) )
57 id 19 . . . . . . . . . . . 12  |-  ( ( x C M ) 
C_  ( ( `' g " { c } )  i^i  (
s C M ) )  ->  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) )
58 inss1 3477 . . . . . . . . . . . 12  |-  ( ( `' g " {
c } )  i^i  ( s C M ) )  C_  ( `' g " {
c } )
5957, 58syl6ss 3277 . . . . . . . . . . 11  |-  ( ( x C M ) 
C_  ( ( `' g " { c } )  i^i  (
s C M ) )  ->  ( x C M )  C_  ( `' g " {
c } ) )
6059a1i 10 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
( x C M )  C_  ( ( `' g " {
c } )  i^i  ( s C M ) )  ->  (
x C M ) 
C_  ( `' g
" { c } ) ) )
6160anim2d 548 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
( ( F `  c )  <_  ( # `
 x )  /\  ( x C M )  C_  ( ( `' g " {
c } )  i^i  ( s C M ) ) )  -> 
( ( F `  c )  <_  ( # `
 x )  /\  ( x C M )  C_  ( `' g " { c } ) ) ) )
6256, 61anim12d 546 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
( x  e.  ~P s  /\  ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) ) )  ->  ( x  e.  ~P t  /\  (
( F `  c
)  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' g " {
c } ) ) ) ) )
6362reximdv2 2737 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( E. x  e.  ~P  s ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) )  ->  E. x  e.  ~P  t ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  ( `' g " {
c } ) ) ) )
6463reximdv 2739 . . . . . 6  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( E. c  e.  R  E. x  e.  ~P  s ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) )  ->  E. c  e.  R  E. x  e.  ~P  t ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  ( `' g " {
c } ) ) ) )
6549, 64mpd 14 . . . . 5  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  E. c  e.  R  E. x  e.  ~P  t ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' g " {
c } ) ) )
6665ex 423 . . . 4  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( ( ( 1 ... N ) 
~~  s  /\  s  C_  t )  ->  E. c  e.  R  E. x  e.  ~P  t ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' g " {
c } ) ) ) )
6766exlimdv 1641 . . 3  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( E. s
( ( 1 ... N )  ~~  s  /\  s  C_  t )  ->  E. c  e.  R  E. x  e.  ~P  t ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  ( `' g " {
c } ) ) ) )
6817, 67mpd 14 . 2  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  E. c  e.  R  E. x  e.  ~P  t ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  ( `' g " {
c } ) ) )
691, 2, 3, 4, 5, 68ramub 13268 1  |-  ( ph  ->  ( M Ramsey  F )  <_  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1546    = wceq 1647    e. wcel 1715   E.wrex 2629   {crab 2632   _Vcvv 2873    i^i cin 3237    C_ wss 3238   ~Pcpw 3714   {csn 3729   class class class wbr 4125   `'ccnv 4791    |` cres 4794   "cima 4795   -->wf 5354   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983    ~~ cen 7003    ~<_ cdom 7004   Fincfn 7006   1c1 8885    <_ cle 9015   NN0cn0 10114   ...cfz 10935   #chash 11505   Ramsey cram 13254
This theorem is referenced by:  ramub1  13283
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-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
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-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  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-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-hash 11506  df-ram 13256
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