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.

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 12109	. . . . . . 7 |- ( N e. NN0 -> ( # ` ( 1 ... N ) ) = N )
8	6, 7	syl 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 ) )
10	8, 9	eqbrtrd 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 ) )
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 210	. . . 4 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> ( 1 ... N ) ~<_ t )
16	12	domen 7315	. . . 4 |- ( ( 1 ... N ) ~<_ t <-> E. s ( ( 1 ... N ) ~~ s /\ s C_ t ) )
17	15, 16	sylib 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 ) )
20	19	ad2antrl 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 ) ) )
22	20, 21	syl 16	. . . . . 6 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> ( # ` s ) = ( # ` ( 1 ... N ) ) )
23	5	ad2antrr 725	. . . . . . 7 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> N e. NN0 )
24	23, 7	syl 16	. . . . . 6 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> ( # ` ( 1 ... N ) ) = N )
25	22, 24	eqtrd 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 )
27	12	a1i 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 )
29	2	ad2antrr 725	. . . . . . 7 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> M e. NN0 )
30	1	hashbcss 14057	. . . . . . 7 |- ( ( t e. _V /\ s C_ t /\ M e. NN0 ) -> ( s C M ) C_ ( t C M ) )
31	27, 28, 29, 30	syl3anc 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 )
33	26, 31, 32	syl2anc 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
35	34	resex 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 ) )
37	36	anbi2d 703	. . . . . . . 8 |- ( f = ( g |` ( s C M ) ) -> ( ( ( # ` s ) = N /\ f : ( s C M ) --> R ) <-> ( ( # ` s ) = N /\ ( g |` ( s C M ) ) : ( s C M ) --> R ) ) )
38	37	anbi2d 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 ) ) )
40	39	imaeq1d 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 ) )
42	40, 41	syl6eq 2486	. . . . . . . . . 10 |- ( f = ( g |` ( s C M ) ) -> ( `' f " { c } ) = ( ( `' g " { c } ) i^i ( s C M ) ) )
43	42	sseq2d 3379	. . . . . . . . 9 |- ( f = ( g |` ( s C M ) ) -> ( ( x C M ) C_ ( `' f " { c } ) <-> ( x C M ) C_ ( ( `' g " { c } ) i^i ( s C M ) ) ) )
44	43	anbi2d 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 ) ) ) ) )
45	44	2rexbidv 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 ) ) ) ) )
46	38, 45	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 ) ) ) ) ) )
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 } ) ) )
48	35, 46, 47	vtocl 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 ) ) ) )
49	18, 25, 33, 48	syl12anc 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 )
51	50	expcom 435	. . . . . . . . 9 |- ( s C_ t -> ( x C_ s -> x C_ t ) )
52	51	ad2antll 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 )
55	52, 53, 54	3imtr4g 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 } )
58	56, 57	syl6ss 3363	. . . . . . . . 9 |- ( ( x C M ) C_ ( ( `' g " { c } ) i^i ( s C M ) ) -> ( x C M ) C_ ( `' g " { c } ) )
59	58	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 } ) ) )
60	59	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 } ) ) ) )
61	55, 60	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 } ) ) ) ) )
62	61	reximdv2 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 } ) ) ) )
63	62	reximdv 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 } ) ) ) )
64	49, 63	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 } ) ) )
65	17, 64	exlimddv 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 } ) ) )
66	1, 2, 3, 4, 5, 65	ramub 14066	1 |- ( ph -> ( M Ramsey F ) <_ N )

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