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Theorem ramub2 14743
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 12468 . . . . . . 7  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
86, 7syl 17 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( # `  (
1 ... N ) )  =  N )
9 simprl 758 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  N  <_  ( # `
 t ) )
108, 9eqbrtrd 4417 . . . . 5  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( # `  (
1 ... N ) )  <_  ( # `  t
) )
11 fzfid 12126 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( 1 ... N )  e.  Fin )
12 vex 3064 . . . . . 6  |-  t  e. 
_V
13 hashdom 12497 . . . . . 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 212 . . . 4  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( 1 ... N )  ~<_  t )
1612domen 7569 . . . 4  |-  ( ( 1 ... N )  ~<_  t  <->  E. s ( ( 1 ... N ) 
~~  s  /\  s  C_  t ) )
1715, 16sylib 198 . . 3  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  E. s ( ( 1 ... N ) 
~~  s  /\  s  C_  t ) )
18 simpll 754 . . . . 5  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ph )
19 ensym 7604 . . . . . . . 8  |-  ( ( 1 ... N ) 
~~  s  ->  s  ~~  ( 1 ... N
) )
2019ad2antrl 728 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  s  ~~  ( 1 ... N
) )
21 hasheni 12470 . . . . . . 7  |-  ( s 
~~  ( 1 ... N )  ->  ( # `
 s )  =  ( # `  (
1 ... N ) ) )
2220, 21syl 17 . . . . . 6  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( # `
 s )  =  ( # `  (
1 ... N ) ) )
235ad2antrr 726 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  N  e.  NN0 )
2423, 7syl 17 . . . . . 6  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( # `
 ( 1 ... N ) )  =  N )
2522, 24eqtrd 2445 . . . . 5  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( # `
 s )  =  N )
26 simplrr 765 . . . . . 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 760 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  s  C_  t )
292ad2antrr 726 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  M  e.  NN0 )
301hashbcss 14733 . . . . . . 7  |-  ( ( t  e.  _V  /\  s  C_  t  /\  M  e.  NN0 )  ->  (
s C M ) 
C_  ( t C M ) )
3127, 28, 29, 30syl3anc 1232 . . . . . 6  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
s C M ) 
C_  ( t C M ) )
3226, 31fssresd 5737 . . . . 5  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
g  |`  ( s C M ) ) : ( s C M ) --> R )
33 vex 3064 . . . . . . 7  |-  g  e. 
_V
3433resex 5139 . . . . . 6  |-  ( g  |`  ( s C M ) )  e.  _V
35 feq1 5698 . . . . . . . . 9  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
f : ( s C M ) --> R  <-> 
( g  |`  (
s C M ) ) : ( s C M ) --> R ) )
3635anbi2d 704 . . . . . . . 8  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
( ( # `  s
)  =  N  /\  f : ( s C M ) --> R )  <-> 
( ( # `  s
)  =  N  /\  ( g  |`  (
s C M ) ) : ( s C M ) --> R ) ) )
3736anbi2d 704 . . . . . . 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 ) ) ) )
38 cnveq 4999 . . . . . . . . . . . 12  |-  ( f  =  ( g  |`  ( s C M ) )  ->  `' f  =  `' (
g  |`  ( s C M ) ) )
3938imaeq1d 5158 . . . . . . . . . . 11  |-  ( f  =  ( g  |`  ( s C M ) )  ->  ( `' f " {
c } )  =  ( `' ( g  |`  ( s C M ) ) " {
c } ) )
40 cnvresima 5314 . . . . . . . . . . 11  |-  ( `' ( g  |`  (
s C M ) ) " { c } )  =  ( ( `' g " { c } )  i^i  ( s C M ) )
4139, 40syl6eq 2461 . . . . . . . . . 10  |-  ( f  =  ( g  |`  ( s C M ) )  ->  ( `' f " {
c } )  =  ( ( `' g
" { c } )  i^i  ( s C M ) ) )
4241sseq2d 3472 . . . . . . . . 9  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
( x C M )  C_  ( `' f " { c } )  <->  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) ) )
4342anbi2d 704 . . . . . . . 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 ) ) ) ) )
44432rexbidv 2927 . . . . . . 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 ) ) ) ) )
4537, 44imbi12d 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 ) ) ) ) ) )
46 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 } ) ) )
4734, 45, 46vtocl 3113 . . . . 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 ) ) ) )
4818, 25, 32, 47syl12anc 1230 . . . 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 ) ) ) )
49 sstr 3452 . . . . . . . . . 10  |-  ( ( x  C_  s  /\  s  C_  t )  ->  x  C_  t )
5049expcom 435 . . . . . . . . 9  |-  ( s 
C_  t  ->  (
x  C_  s  ->  x 
C_  t ) )
5150ad2antll 729 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
x  C_  s  ->  x 
C_  t ) )
52 selpw 3964 . . . . . . . 8  |-  ( x  e.  ~P s  <->  x  C_  s
)
53 selpw 3964 . . . . . . . 8  |-  ( x  e.  ~P t  <->  x  C_  t
)
5451, 52, 533imtr4g 272 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
x  e.  ~P s  ->  x  e.  ~P t
) )
55 id 23 . . . . . . . . . 10  |-  ( ( x C M ) 
C_  ( ( `' g " { c } )  i^i  (
s C M ) )  ->  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) )
56 inss1 3661 . . . . . . . . . 10  |-  ( ( `' g " {
c } )  i^i  ( s C M ) )  C_  ( `' g " {
c } )
5755, 56syl6ss 3456 . . . . . . . . 9  |-  ( ( x C M ) 
C_  ( ( `' g " { c } )  i^i  (
s C M ) )  ->  ( x C M )  C_  ( `' g " {
c } ) )
5857a1i 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 } ) ) )
5958anim2d 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 } ) ) ) )
6054, 59anim12d 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 } ) ) ) ) )
6160reximdv2 2877 . . . . 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 } ) ) ) )
6261reximdv 2880 . . . 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 } ) ) ) )
6348, 62mpd 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 } ) ) )
6417, 63exlimddv 1749 . 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 } ) ) )
651, 2, 3, 4, 5, 64ramub 14742 1  |-  ( ph  ->  ( M Ramsey  F )  <_  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407   E.wex 1635    e. wcel 1844   E.wrex 2757   {crab 2760   _Vcvv 3061    i^i cin 3415    C_ wss 3416   ~Pcpw 3957   {csn 3974   class class class wbr 4397   `'ccnv 4824    |` cres 4827   "cima 4828   -->wf 5567   ` cfv 5571  (class class class)co 6280    |-> cmpt2 6282    ~~ cen 7553    ~<_ cdom 7554   Fincfn 7556   1c1 9525    <_ cle 9661   NN0cn0 10838   ...cfz 11728   #chash 12454   Ramsey cram 14728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-card 8354  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-hash 12455  df-ram 14730
This theorem is referenced by:  ramub1  14757
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