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.

Step	Hyp	Ref	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 )
6	5	adantr 465	. . . . . . 7 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> N e. NN0 )
7		hashfz1 12468	. . . . . . 7 |- ( N e. NN0 -> ( # ` ( 1 ... N ) ) = N )
8	6, 7	syl 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 ) )
10	8, 9	eqbrtrd 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 ) )
14	11, 12, 13	sylancl 662	. . . . 5 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> ( ( # ` ( 1 ... N ) ) <_ ( # ` t ) <-> ( 1 ... N ) ~<_ t ) )
15	10, 14	mpbid 212	. . . 4 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> ( 1 ... N ) ~<_ t )
16	12	domen 7569	. . . 4 |- ( ( 1 ... N ) ~<_ t <-> E. s ( ( 1 ... N ) ~~ s /\ s C_ t ) )
17	15, 16	sylib 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 ) )
20	19	ad2antrl 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 ) ) )
22	20, 21	syl 17	. . . . . 6 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> ( # ` s ) = ( # ` ( 1 ... N ) ) )
23	5	ad2antrr 726	. . . . . . 7 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> N e. NN0 )
24	23, 7	syl 17	. . . . . 6 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> ( # ` ( 1 ... N ) ) = N )
25	22, 24	eqtrd 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 )
27	12	a1i 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 )
29	2	ad2antrr 726	. . . . . . 7 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> M e. NN0 )
30	1	hashbcss 14733	. . . . . . 7 |- ( ( t e. _V /\ s C_ t /\ M e. NN0 ) -> ( s C M ) C_ ( t C M ) )
31	27, 28, 29, 30	syl3anc 1232	. . . . . 6 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> ( s C M ) C_ ( t C M ) )
32	26, 31	fssresd 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
34	33	resex 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 ) )
36	35	anbi2d 704	. . . . . . . 8 |- ( f = ( g |` ( s C M ) ) -> ( ( ( # ` s ) = N /\ f : ( s C M ) --> R ) <-> ( ( # ` s ) = N /\ ( g |` ( s C M ) ) : ( s C M ) --> R ) ) )
37	36	anbi2d 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 ) ) )
39	38	imaeq1d 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 ) )
41	39, 40	syl6eq 2461	. . . . . . . . . 10 |- ( f = ( g |` ( s C M ) ) -> ( `' f " { c } ) = ( ( `' g " { c } ) i^i ( s C M ) ) )
42	41	sseq2d 3472	. . . . . . . . 9 |- ( f = ( g |` ( s C M ) ) -> ( ( x C M ) C_ ( `' f " { c } ) <-> ( x C M ) C_ ( ( `' g " { c } ) i^i ( s C M ) ) ) )
43	42	anbi2d 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 ) ) ) ) )
44	43	2rexbidv 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 ) ) ) ) )
45	37, 44	imbi12d 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 } ) ) )
47	34, 45, 46	vtocl 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 ) ) ) )
48	18, 25, 32, 47	syl12anc 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 )
50	49	expcom 435	. . . . . . . . 9 |- ( s C_ t -> ( x C_ s -> x C_ t ) )
51	50	ad2antll 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 )
54	51, 52, 53	3imtr4g 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 } )
57	55, 56	syl6ss 3456	. . . . . . . . 9 |- ( ( x C M ) C_ ( ( `' g " { c } ) i^i ( s C M ) ) -> ( x C M ) C_ ( `' g " { c } ) )
58	57	a1i 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 } ) ) )
59	58	anim2d 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 } ) ) ) )
60	54, 59	anim12d 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 } ) ) ) ) )
61	60	reximdv2 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 } ) ) ) )
62	61	reximdv 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 } ) ) ) )
63	48, 62	mpd 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 } ) ) )
64	17, 63	exlimddv 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 } ) ) )
65	1, 2, 3, 4, 5, 64	ramub 14742	1 |- ( ph -> ( M Ramsey F ) <_ N )

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