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Theorem ramub2 15050
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 472 . . . . . . 7  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  N  e.  NN0 )
7 hashfz1 12567 . . . . . . 7  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
86, 7syl 17 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( # `  (
1 ... N ) )  =  N )
9 simprl 772 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  N  <_  ( # `
 t ) )
108, 9eqbrtrd 4416 . . . . 5  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( # `  (
1 ... N ) )  <_  ( # `  t
) )
11 fzfid 12224 . . . . . 6  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( 1 ... N )  e.  Fin )
12 vex 3034 . . . . . 6  |-  t  e. 
_V
13 hashdom 12596 . . . . . 6  |-  ( ( ( 1 ... N
)  e.  Fin  /\  t  e.  _V )  ->  ( ( # `  (
1 ... N ) )  <_  ( # `  t
)  <->  ( 1 ... N )  ~<_  t ) )
1411, 12, 13sylancl 675 . . . . 5  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( ( # `  ( 1 ... N
) )  <_  ( # `
 t )  <->  ( 1 ... N )  ~<_  t ) )
1510, 14mpbid 215 . . . 4  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  ( 1 ... N )  ~<_  t )
1612domen 7600 . . . 4  |-  ( ( 1 ... N )  ~<_  t  <->  E. s ( ( 1 ... N ) 
~~  s  /\  s  C_  t ) )
1715, 16sylib 201 . . 3  |-  ( (
ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  ->  E. s ( ( 1 ... N ) 
~~  s  /\  s  C_  t ) )
18 simpll 768 . . . . 5  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ph )
19 ensym 7636 . . . . . . . 8  |-  ( ( 1 ... N ) 
~~  s  ->  s  ~~  ( 1 ... N
) )
2019ad2antrl 742 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  s  ~~  ( 1 ... N
) )
21 hasheni 12569 . . . . . . 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 740 . . . . . . 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 2505 . . . . 5  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  ( # `
 s )  =  N )
26 simplrr 779 . . . . . 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 774 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  s  C_  t )
292ad2antrr 740 . . . . . . 7  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  M  e.  NN0 )
301hashbcss 15035 . . . . . . 7  |-  ( ( t  e.  _V  /\  s  C_  t  /\  M  e.  NN0 )  ->  (
s C M ) 
C_  ( t C M ) )
3127, 28, 29, 30syl3anc 1292 . . . . . 6  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
s C M ) 
C_  ( t C M ) )
3226, 31fssresd 5762 . . . . 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 3034 . . . . . . 7  |-  g  e. 
_V
3433resex 5154 . . . . . 6  |-  ( g  |`  ( s C M ) )  e.  _V
35 feq1 5720 . . . . . . . . 9  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
f : ( s C M ) --> R  <-> 
( g  |`  (
s C M ) ) : ( s C M ) --> R ) )
3635anbi2d 718 . . . . . . . 8  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
( ( # `  s
)  =  N  /\  f : ( s C M ) --> R )  <-> 
( ( # `  s
)  =  N  /\  ( g  |`  (
s C M ) ) : ( s C M ) --> R ) ) )
3736anbi2d 718 . . . . . . 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 5013 . . . . . . . . . . . 12  |-  ( f  =  ( g  |`  ( s C M ) )  ->  `' f  =  `' (
g  |`  ( s C M ) ) )
3938imaeq1d 5173 . . . . . . . . . . 11  |-  ( f  =  ( g  |`  ( s C M ) )  ->  ( `' f " {
c } )  =  ( `' ( g  |`  ( s C M ) ) " {
c } ) )
40 cnvresima 5331 . . . . . . . . . . 11  |-  ( `' ( g  |`  (
s C M ) ) " { c } )  =  ( ( `' g " { c } )  i^i  ( s C M ) )
4139, 40syl6eq 2521 . . . . . . . . . 10  |-  ( f  =  ( g  |`  ( s C M ) )  ->  ( `' f " {
c } )  =  ( ( `' g
" { c } )  i^i  ( s C M ) ) )
4241sseq2d 3446 . . . . . . . . 9  |-  ( f  =  ( g  |`  ( s C M ) )  ->  (
( x C M )  C_  ( `' f " { c } )  <->  ( x C M )  C_  (
( `' g " { c } )  i^i  ( s C M ) ) ) )
4342anbi2d 718 . . . . . . . 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 2897 . . . . . . 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 327 . . . . . 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 3086 . . . . 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 1290 . . . 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 3426 . . . . . . . . . 10  |-  ( ( x  C_  s  /\  s  C_  t )  ->  x  C_  t )
5049expcom 442 . . . . . . . . 9  |-  ( s 
C_  t  ->  (
x  C_  s  ->  x 
C_  t ) )
5150ad2antll 743 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  <_  ( # `  t
)  /\  g :
( t C M ) --> R ) )  /\  ( ( 1 ... N )  ~~  s  /\  s  C_  t
) )  ->  (
x  C_  s  ->  x 
C_  t ) )
52 selpw 3949 . . . . . . . 8  |-  ( x  e.  ~P s  <->  x  C_  s
)
53 selpw 3949 . . . . . . . 8  |-  ( x  e.  ~P t  <->  x  C_  t
)
5451, 52, 533imtr4g 278 . . . . . . 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 22 . . . . . . . . . 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 3643 . . . . . . . . . 10  |-  ( ( `' g " {
c } )  i^i  ( s C M ) )  C_  ( `' g " {
c } )
5755, 56syl6ss 3430 . . . . . . . . 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 575 . . . . . . 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 572 . . . . . 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 2855 . . . . 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 2857 . . . 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 1789 . 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 15049 1  |-  ( ph  ->  ( M Ramsey  F )  <_  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   E.wrex 2757   {crab 2760   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   {csn 3959   class class class wbr 4395   `'ccnv 4838    |` cres 4841   "cima 4842   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310    ~~ cen 7584    ~<_ cdom 7585   Fincfn 7587   1c1 9558    <_ cle 9694   NN0cn0 10893   ...cfz 11810   #chash 12553   Ramsey cram 15028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-ram 15031
This theorem is referenced by:  ramub1  15065
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