Theorem ramub2 15718

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 481	. . . . . . 7 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> N e. NN0 )
7		hashfz1 13134	. . . . . . 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 794	. . . . . 6 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> N <_ ( # ` t ) )
10	8, 9	eqbrtrd 4675	. . . . 5 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> ( # ` ( 1 ... N ) ) <_ ( # ` t ) )
11		fzfid 12772	. . . . . 6 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> ( 1 ... N ) e. Fin )
12		vex 3203	. . . . . 6 |- t e. _V
13		hashdom 13168	. . . . . 6 |- ( ( ( 1 ... N ) e. Fin /\ t e. _V ) -> ( ( # ` ( 1 ... N ) ) <_ ( # ` t ) <-> ( 1 ... N ) ~<_ t ) )
14	11, 12, 13	sylancl 694	. . . . 5 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> ( ( # ` ( 1 ... N ) ) <_ ( # ` t ) <-> ( 1 ... N ) ~<_ t ) )
15	10, 14	mpbid 222	. . . 4 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> ( 1 ... N ) ~<_ t )
16	12	domen 7968	. . . 4 |- ( ( 1 ... N ) ~<_ t <-> E. s ( ( 1 ... N ) ~~ s /\ s C_ t ) )
17	15, 16	sylib 208	. . 3 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> E. s ( ( 1 ... N ) ~~ s /\ s C_ t ) )
18		simpll 790	. . . . 5 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> ph )
19		ensym 8005	. . . . . . . 8 |- ( ( 1 ... N ) ~~ s -> s ~~ ( 1 ... N ) )
20	19	ad2antrl 764	. . . . . . 7 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> s ~~ ( 1 ... N ) )
21		hasheni 13136	. . . . . . 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 762	. . . . . . 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 2656	. . . . 5 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> ( # ` s ) = N )
26		simplrr 801	. . . . . 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 796	. . . . . . 7 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> s C_ t )
29	2	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> M e. NN0 )
30	1	hashbcss 15708	. . . . . . 7 |- ( ( t e. _V /\ s C_ t /\ M e. NN0 ) -> ( s C M ) C_ ( t C M ) )
31	27, 28, 29, 30	syl3anc 1326	. . . . . 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 6071	. . . . 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 3203	. . . . . . 7 |- g e. _V
34	33	resex 5443	. . . . . 6 |- ( g |` ( s C M ) ) e. _V
35		feq1 6026	. . . . . . . . 9 |- ( f = ( g |` ( s C M ) ) -> ( f : ( s C M ) --> R <-> ( g |` ( s C M ) ) : ( s C M ) --> R ) )
36	35	anbi2d 740	. . . . . . . 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 740	. . . . . . 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 5296	. . . . . . . . . . . 12 |- ( f = ( g |` ( s C M ) ) -> `' f = `' ( g |` ( s C M ) ) )
39	38	imaeq1d 5465	. . . . . . . . . . 11 |- ( f = ( g |` ( s C M ) ) -> ( `' f " { c } ) = ( `' ( g |` ( s C M ) ) " { c } ) )
40		cnvresima 5623	. . . . . . . . . . 11 |- ( `' ( g |` ( s C M ) ) " { c } ) = ( ( `' g " { c } ) i^i ( s C M ) )
41	39, 40	syl6eq 2672	. . . . . . . . . 10 |- ( f = ( g |` ( s C M ) ) -> ( `' f " { c } ) = ( ( `' g " { c } ) i^i ( s C M ) ) )
42	41	sseq2d 3633	. . . . . . . . 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 740	. . . . . . . 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 3057	. . . . . . 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 334	. . . . . 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 3259	. . . . 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 1324	. . . 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 3611	. . . . . . . . . 10 |- ( ( x C_ s /\ s C_ t ) -> x C_ t )
50	49	expcom 451	. . . . . . . . 9 |- ( s C_ t -> ( x C_ s -> x C_ t ) )
51	50	ad2antll 765	. . . . . . . 8 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> ( x C_ s -> x C_ t ) )
52		selpw 4165	. . . . . . . 8 |- ( x e. ~P s <-> x C_ s )
53		selpw 4165	. . . . . . . 8 |- ( x e. ~P t <-> x C_ t )
54	51, 52, 53	3imtr4g 285	. . . . . . 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 3833	. . . . . . . . . 10 |- ( ( `' g " { c } ) i^i ( s C M ) ) C_ ( `' g " { c } )
57	55, 56	syl6ss 3615	. . . . . . . . 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 589	. . . . . . 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 586	. . . . . 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 3014	. . . . 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 3016	. . . 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 1863	. 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 15717	1 |- ( ph -> ( M Ramsey F ) <_ N )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   class class class wbr 4653   `'ccnv 5113   |` cres 5116   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955   1c1 9937   <_ cle 10075   NN0cn0 11292   ...cfz 12326   #chash 13117   Ramsey cram 15703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-ram 15705

This theorem is referenced by:  ramub1  15732