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Theorem ramub2 14067
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 465 . . . . . . 7  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  N  e.  NN0 )
7 hashfz1 12109 . . . . . . 7  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
86, 7syl 16 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( # `  (
1 ... N ) )  =  N )
9 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  N  <_  ( # `
 t ) )
108, 9eqbrtrd 4307 . . . . 5  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( # `  (
1 ... N ) )  <_  ( # `  t
) )
11 fzfid 11787 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( 1 ... N )  e.  Fin )
12 vex 2970 . . . . . 6  |-  t  e. 
_V
13 hashdom 12134 . . . . . 6  |-  ( ( ( 1 ... N
)  e.  Fin  /\  t  e.  _V )  ->  ( ( # `  (
1 ... N ) )  <_  ( # `  t
)  <->  ( 1 ... N )  ~<_  t ) )
1411, 12, 13sylancl 662 . . . . 5  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( ( # `  ( 1 ... N
) )  <_  ( # `
 t )  <->  ( 1 ... N )  ~<_  t ) )
1510, 14mpbid 210 . . . 4  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( 1 ... N )  ~<_  t )
1612domen 7315 . . . 4  |-  ( ( 1 ... N )  ~<_  t  <->  E. s ( ( 1 ... N ) 
~~  s  /\  s  C_  t ) )
1715, 16sylib 196 . . 3  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  E. s ( ( 1 ... N ) 
~~  s  /\  s  C_  t ) )
18 simpll 753 . . . . 5  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ph )
19 ensym 7350 . . . . . . . 8  |-  ( ( 1 ... N ) 
~~  s  ->  s  ~~  ( 1 ... N
) )
2019ad2antrl 727 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  s  ~~  ( 1 ... N
) )
21 hasheni 12111 . . . . . . 7  |-  ( s 
~~  ( 1 ... N )  ->  ( # `
 s )  =  ( # `  (
1 ... N ) ) )
2220, 21syl 16 . . . . . 6  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( # `
 s )  =  ( # `  (
1 ... N ) ) )
235ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  N  e.  NN0 )
2423, 7syl 16 . . . . . 6  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( # `
 ( 1 ... N ) )  =  N )
2522, 24eqtrd 2470 . . . . 5  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( # `
 s )  =  N )
26 simplrr 760 . . . . . 6  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  g : ( t C M ) --> R )
2712a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  t  e.  _V )
28 simprr 756 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  s  C_  t )
292ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  M  e.  NN0 )
301hashbcss 14057 . . . . . . 7  |-  ( ( t  e.  _V  /\  s  C_  t  /\  M  e.  NN0 )  ->  (
s C M ) 
C_  ( t C M ) )
3127, 28, 29, 30syl3anc 1218 . . . . . 6  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
s C M ) 
C_  ( t C M ) )
32 fssres 5573 . . . . . 6  |-  ( ( g : ( t C M ) --> R  /\  ( s C M )  C_  (
t C M ) )  ->  ( g  |`  ( s C M ) ) : ( s C M ) --> R )
3326, 31, 32syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
g  |`  ( s C M ) ) : ( s C M ) --> R )
34 vex 2970 . . . . . . 7  |-  g  e. 
_V
3534resex 5145 . . . . . 6  |-  ( g  |`  ( s C M ) )  e.  _V
36 feq1 5537 . . . . . . . . 9  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
f : ( s C M ) --> R  <-> 
( g  |`  (
s C M ) ) : ( s C M ) --> R ) )
3736anbi2d 703 . . . . . . . 8  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
( ( # `  s
)  =  N  /\  f : ( s C M ) --> R )  <-> 
( ( # `  s
)  =  N  /\  ( g  |`  (
s C M ) ) : ( s C M ) --> R ) ) )
3837anbi2d 703 . . . . . . 7  |-  ( 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 5008 . . . . . . . . . . . 12  |-  ( f  =  ( g  |`  ( s C M ) )  ->  `' f  =  `' (
g  |`  ( s C M ) ) )
4039imaeq1d 5163 . . . . . . . . . . 11  |-  ( f  =  ( g  |`  ( s C M ) )  ->  ( `' f " {
c } )  =  ( `' ( g  |`  ( s C M ) ) " {
c } ) )
41 cnvresima 5322 . . . . . . . . . . 11  |-  ( `' ( g  |`  (
s C M ) ) " { c } )  =  ( ( `' g " { c } )  i^i  ( s C M ) )
4240, 41syl6eq 2486 . . . . . . . . . 10  |-  ( f  =  ( g  |`  ( s C M ) )  ->  ( `' f " {
c } )  =  ( ( `' g
" { c } )  i^i  ( s C M ) ) )
4342sseq2d 3379 . . . . . . . . 9  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
( x C M )  C_  ( `' f " { c } )  <->  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) ) )
4443anbi2d 703 . . . . . . . 8  |-  ( 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 2753 . . . . . . 7  |-  ( 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 320 . . . . . 6  |-  ( 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 . . . . . 6  |-  ( (
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 3019 . . . . 5  |-  ( (
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 1216 . . . 4  |-  ( ( ( 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 3359 . . . . . . . . . 10  |-  ( ( x  C_  s  /\  s  C_  t )  ->  x  C_  t )
5150expcom 435 . . . . . . . . 9  |-  ( s 
C_  t  ->  (
x  C_  s  ->  x 
C_  t ) )
5251ad2antll 728 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
x  C_  s  ->  x 
C_  t ) )
53 selpw 3862 . . . . . . . 8  |-  ( x  e.  ~P s  <->  x  C_  s
)
54 selpw 3862 . . . . . . . 8  |-  ( x  e.  ~P t  <->  x  C_  t
)
5552, 53, 543imtr4g 270 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
x  e.  ~P s  ->  x  e.  ~P t
) )
56 id 22 . . . . . . . . . 10  |-  ( ( x C M ) 
C_  ( ( `' g " { c } )  i^i  (
s C M ) )  ->  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) )
57 inss1 3565 . . . . . . . . . 10  |-  ( ( `' g " {
c } )  i^i  ( s C M ) )  C_  ( `' g " {
c } )
5856, 57syl6ss 3363 . . . . . . . . 9  |-  ( ( x C M ) 
C_  ( ( `' g " { c } )  i^i  (
s C M ) )  ->  ( x C M )  C_  ( `' g " {
c } ) )
5958a1i 11 . . . . . . . 8  |-  ( ( ( 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 } ) ) )
6059anim2d 565 . . . . . . 7  |-  ( ( ( 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 } ) ) ) )
6155, 60anim12d 563 . . . . . 6  |-  ( ( ( 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 } ) ) ) ) )
6261reximdv2 2820 . . . . 5  |-  ( ( ( 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 } ) ) ) )
6362reximdv 2822 . . . 4  |-  ( ( ( 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 } ) ) ) )
6449, 63mpd 15 . . 3  |-  ( ( ( 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 } ) ) )
6517, 64exlimddv 1692 . 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 } ) ) )
661, 2, 3, 4, 5, 65ramub 14066 1  |-  ( ph  ->  ( M Ramsey  F )  <_  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   E.wrex 2711   {crab 2714   _Vcvv 2967    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   {csn 3872   class class class wbr 4287   `'ccnv 4834    |` cres 4837   "cima 4838   -->wf 5409   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088    ~~ cen 7299    ~<_ cdom 7300   Fincfn 7302   1c1 9275    <_ cle 9411   NN0cn0 10571   ...cfz 11429   #chash 12095   Ramsey cram 14052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-hash 12096  df-ram 14054
This theorem is referenced by:  ramub1  14081
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