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.

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 472	. . . . . . 7 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> N e. NN0 )
7		hashfz1 12567	. . . . . . 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 772	. . . . . 6 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> N <_ ( # ` t ) )
10	8, 9	eqbrtrd 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 ) )
14	11, 12, 13	sylancl 675	. . . . 5 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> ( ( # ` ( 1 ... N ) ) <_ ( # ` t ) <-> ( 1 ... N ) ~<_ t ) )
15	10, 14	mpbid 215	. . . 4 |- ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) -> ( 1 ... N ) ~<_ t )
16	12	domen 7600	. . . 4 |- ( ( 1 ... N ) ~<_ t <-> E. s ( ( 1 ... N ) ~~ s /\ s C_ t ) )
17	15, 16	sylib 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 ) )
20	19	ad2antrl 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 ) ) )
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 740	. . . . . . 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 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 )
27	12	a1i 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 )
29	2	ad2antrr 740	. . . . . . 7 |- ( ( ( ph /\ ( N <_ ( # ` t ) /\ g : ( t C M ) --> R ) ) /\ ( ( 1 ... N ) ~~ s /\ s C_ t ) ) -> M e. NN0 )
30	1	hashbcss 15035	. . . . . . 7 |- ( ( t e. _V /\ s C_ t /\ M e. NN0 ) -> ( s C M ) C_ ( t C M ) )
31	27, 28, 29, 30	syl3anc 1292	. . . . . 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 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
34	33	resex 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 ) )
36	35	anbi2d 718	. . . . . . . 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 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 ) ) )
39	38	imaeq1d 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 ) )
41	39, 40	syl6eq 2521	. . . . . . . . . 10 |- ( f = ( g |` ( s C M ) ) -> ( `' f " { c } ) = ( ( `' g " { c } ) i^i ( s C M ) ) )
42	41	sseq2d 3446	. . . . . . . . 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 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 ) ) ) ) )
44	43	2rexbidv 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 ) ) ) ) )
45	37, 44	imbi12d 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 } ) ) )
47	34, 45, 46	vtocl 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 ) ) ) )
48	18, 25, 32, 47	syl12anc 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 )
50	49	expcom 442	. . . . . . . . 9 |- ( s C_ t -> ( x C_ s -> x C_ t ) )
51	50	ad2antll 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 )
54	51, 52, 53	3imtr4g 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 } )
57	55, 56	syl6ss 3430	. . . . . . . . 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 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 } ) ) ) )
60	54, 59	anim12d 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 } ) ) ) ) )
61	60	reximdv2 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 } ) ) ) )
62	61	reximdv 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 } ) ) ) )
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 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 } ) ) )
65	1, 2, 3, 4, 5, 64	ramub 15049	1 |- ( ph -> ( M Ramsey F ) <_ N )

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