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Theorem 0ram2 13391
Description: The Ramsey number when  M  = 
0. (Contributed by Mario Carneiro, 22-Apr-2015.)
Assertion
Ref Expression
0ram2  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )

Proof of Theorem 0ram2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frn 5599 . . . . 5  |-  ( F : R --> NN0  ->  ran 
F  C_  NN0 )
213ad2ant3 981 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  NN0 )
3 nn0ssz 10304 . . . 4  |-  NN0  C_  ZZ
42, 3syl6ss 3362 . . 3  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  ZZ )
5 nn0ssre 10227 . . . . 5  |-  NN0  C_  RR
62, 5syl6ss 3362 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  RR )
7 simp1 958 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  e.  Fin )
8 ffn 5593 . . . . . . 7  |-  ( F : R --> NN0  ->  F  Fn  R )
983ad2ant3 981 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  F  Fn  R )
10 dffn4 5661 . . . . . 6  |-  ( F  Fn  R  <->  F : R -onto-> ran  F )
119, 10sylib 190 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  F : R -onto-> ran  F )
12 fofi 7394 . . . . 5  |-  ( ( R  e.  Fin  /\  F : R -onto-> ran  F
)  ->  ran  F  e. 
Fin )
137, 11, 12syl2anc 644 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  e.  Fin )
14 fdm 5597 . . . . . . 7  |-  ( F : R --> NN0  ->  dom 
F  =  R )
15143ad2ant3 981 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =  R )
16 simp2 959 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  =/=  (/) )
1715, 16eqnetrd 2621 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =/=  (/) )
18 dm0rn0 5088 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1918necon3bii 2635 . . . . 5  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
2017, 19sylib 190 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  =/=  (/) )
21 fimaxre 9957 . . . 4  |-  ( ( ran  F  C_  RR  /\ 
ran  F  e.  Fin  /\ 
ran  F  =/=  (/) )  ->  E. x  e.  ran  F A. y  e.  ran  F  y  <_  x )
226, 13, 20, 21syl3anc 1185 . . 3  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ran  F A. y  e.  ran  F  y  <_  x )
23 ssrexv 3410 . . 3  |-  ( ran 
F  C_  ZZ  ->  ( E. x  e.  ran  F A. y  e.  ran  F  y  <_  x  ->  E. x  e.  ZZ  A. y  e.  ran  F  y  <_  x ) )
244, 22, 23sylc 59 . 2  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ZZ  A. y  e. 
ran  F  y  <_  x )
25 0ram 13390 . 2  |-  ( ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R --> NN0 )  /\  E. x  e.  ZZ  A. y  e.  ran  F  y  <_  x )  ->  (
0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
2624, 25mpdan 651 1  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    C_ wss 3322   (/)c0 3630   class class class wbr 4214   dom cdm 4880   ran crn 4881    Fn wfn 5451   -->wf 5452   -onto->wfo 5454  (class class class)co 6083   Fincfn 7111   supcsup 7447   RRcr 8991   0cc0 8992    < clt 9122    <_ cle 9123   NN0cn0 10223   ZZcz 10284   Ramsey cram 13369
This theorem is referenced by:  0ramcl  13393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-hash 11621  df-ram 13371
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