Theorem ramub2 14390

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 12386	. . . . . . 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 4467	. . . . 5 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> ( # ` ( 1 ... N ) ) <_ ( # ` t ) )
11		fzfid 12050	. . . . . 6 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> ( 1 ... N ) e. Fin )
12		vex 3116	. . . . . 6 |- t e. _V
13		hashdom 12414	. . . . . 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 7529	. . . 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 7564	. . . . . . . 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 12388	. . . . . . 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 2508	. . . . 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 14380	. . . . . . 7 |- ( ( t e. _V /\ s C_ t /\ M e. NN0 ) -> ( s C M ) C_ ( t C M ) )
31	27, 28, 29, 30	syl3anc 1228	. . . . . 6 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> ( s C M ) C_ ( t C M ) )
32		fssres 5750	. . . . . 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 3116	. . . . . . 7 |- g e. _V
35	34	resex 5316	. . . . . 6 |- ( g |` ( s C M ) ) e. _V
36		feq1 5712	. . . . . . . . 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 5175	. . . . . . . . . . . 12 |- ( f = ( g |` ( s C M ) ) -> `' f = `' ( g |` ( s C M ) ) )
40	39	imaeq1d 5335	. . . . . . . . . . 11 |- ( f = ( g |` ( s C M ) ) -> ( `' f " { c } ) = ( `' ( g |` ( s C M ) ) " { c } ) )
41		cnvresima 5495	. . . . . . . . . . 11 |- ( `' ( g |` ( s C M ) ) " { c } ) = ( ( `' g " { c } ) i^i ( s C M ) )
42	40, 41	syl6eq 2524	. . . . . . . . . 10 |- ( f = ( g |` ( s C M ) ) -> ( `' f " { c } ) = ( ( `' g " { c } ) i^i ( s C M ) ) )
43	42	sseq2d 3532	. . . . . . . . 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 2980	. . . . . . 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 3165	. . . . 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 1226	. . . 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 3512	. . . . . . . . . 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 4017	. . . . . . . 8 |- ( x e. ~P s <-> x C_ s )
54		selpw 4017	. . . . . . . 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 3718	. . . . . . . . . 10 |- ( ( `' g " { c } ) i^i ( s C M ) ) C_ ( `' g " { c } )
58	56, 57	syl6ss 3516	. . . . . . . . 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 2934	. . . . 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 2937	. . . 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 1702	. 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 14389	1 |- ( ph -> ( M Ramsey F ) <_ N )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   E.wrex 2815   {crab 2818   _Vcvv 3113   i^i cin 3475   C_ wss 3476   ~Pcpw 4010   {csn 4027   class class class wbr 4447   `'ccnv 4998   |` cres 5001   "cima 5002   -->wf 5583   ` cfv 5587  (class class class)co 6283   |-> cmpt2 6285   ~~ cen 7513   ~<_ cdom 7514   Fincfn 7516   1c1 9492   <_ cle 9628   NN0cn0 10794   ...cfz 11671   #chash 12372   Ramsey cram 14375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-hash 12373  df-ram 14377

This theorem is referenced by:  ramub1  14404