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Theorem 0ramcl 14924
Description: Lemma for ramcl 14930: Existence of the Ramsey number when  M  =  0. (Contributed by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
0ramcl  |-  ( ( R  e.  Fin  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )

Proof of Theorem 0ramcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 5689 . . . . . . . 8  |-  ( F : R --> NN0  ->  F  Fn  R )
2 dffn4 5759 . . . . . . . 8  |-  ( F  Fn  R  <->  F : R -onto-> ran  F )
31, 2sylib 199 . . . . . . 7  |-  ( F : R --> NN0  ->  F : R -onto-> ran  F
)
43ad2antlr 731 . . . . . 6  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F : R -onto-> ran  F
)
5 foeq2 5750 . . . . . . 7  |-  ( R  =  (/)  ->  ( F : R -onto-> ran  F  <->  F : (/) -onto-> ran  F ) )
65adantl 467 . . . . . 6  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( F : R -onto-> ran  F  <->  F : (/) -onto-> ran  F
) )
74, 6mpbid 213 . . . . 5  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F : (/) -onto-> ran  F )
8 fo00 5808 . . . . . 6  |-  ( F : (/) -onto-> ran  F  <->  ( F  =  (/)  /\  ran  F  =  (/) ) )
98simplbi 461 . . . . 5  |-  ( F : (/) -onto-> ran  F  ->  F  =  (/) )
107, 9syl 17 . . . 4  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F  =  (/) )
1110oveq2d 6265 . . 3  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( 0 Ramsey  F )  =  ( 0 Ramsey  (/) ) )
12 0nn0 10835 . . . . 5  |-  0  e.  NN0
13 ram0 14923 . . . . 5  |-  ( 0  e.  NN0  ->  ( 0 Ramsey  (/) )  =  0 )
1412, 13ax-mp 5 . . . 4  |-  ( 0 Ramsey  (/) )  =  0
1514, 12eqeltri 2502 . . 3  |-  ( 0 Ramsey  (/) )  e.  NN0
1611, 15syl6eqel 2514 . 2  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( 0 Ramsey  F )  e.  NN0 )
17 0ram2 14922 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
18 frn 5695 . . . . . . 7  |-  ( F : R --> NN0  ->  ran 
F  C_  NN0 )
19183ad2ant3 1028 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  NN0 )
20 nn0ssz 10909 . . . . . . . 8  |-  NN0  C_  ZZ
2119, 20syl6ss 3419 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  ZZ )
22 fdm 5693 . . . . . . . . . 10  |-  ( F : R --> NN0  ->  dom 
F  =  R )
23223ad2ant3 1028 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =  R )
24 simp2 1006 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  =/=  (/) )
2523, 24eqnetrd 2668 . . . . . . . 8  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =/=  (/) )
26 dm0rn0 5013 . . . . . . . . 9  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
2726necon3bii 2653 . . . . . . . 8  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
2825, 27sylib 199 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  =/=  (/) )
29 nn0ssre 10824 . . . . . . . . . 10  |-  NN0  C_  RR
3019, 29syl6ss 3419 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  RR )
31 simp1 1005 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  e.  Fin )
3233ad2ant3 1028 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  F : R -onto-> ran  F )
33 fofi 7813 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  F : R -onto-> ran  F
)  ->  ran  F  e. 
Fin )
3431, 32, 33syl2anc 665 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  e.  Fin )
35 fimaxre 10502 . . . . . . . . 9  |-  ( ( ran  F  C_  RR  /\ 
ran  F  e.  Fin  /\ 
ran  F  =/=  (/) )  ->  E. x  e.  ran  F A. y  e.  ran  F  y  <_  x )
3630, 34, 28, 35syl3anc 1264 . . . . . . . 8  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ran  F A. y  e.  ran  F  y  <_  x )
37 ssrexv 3469 . . . . . . . 8  |-  ( 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 ) )
3821, 36, 37sylc 62 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ZZ  A. y  e. 
ran  F  y  <_  x )
39 suprzcl2 11205 . . . . . . 7  |-  ( ( ran  F  C_  ZZ  /\ 
ran  F  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  ran  F  y  <_  x )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
4021, 28, 38, 39syl3anc 1264 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
4119, 40sseldd 3408 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  sup ( ran  F ,  RR ,  <  )  e.  NN0 )
4217, 41eqeltrd 2506 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  e.  NN0 )
43423expa 1205 . . 3  |-  ( ( ( R  e.  Fin  /\  R  =/=  (/) )  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )
4443an32s 811 . 2  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =/=  (/) )  -> 
( 0 Ramsey  F )  e.  NN0 )
4516, 44pm2.61dane 2688 1  |-  ( ( R  e.  Fin  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715    C_ wss 3379   (/)c0 3704   class class class wbr 4366   dom cdm 4796   ran crn 4797    Fn wfn 5539   -->wf 5540   -onto->wfo 5542  (class class class)co 6249   Fincfn 7524   supcsup 7907   RRcr 9489   0cc0 9490    < clt 9626    <_ cle 9627   NN0cn0 10820   ZZcz 10888   Ramsey cram 14892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-2o 7138  df-oadd 7141  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-sup 7909  df-inf 7910  df-card 8325  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-n0 10821  df-z 10889  df-uz 11111  df-rp 11254  df-fz 11736  df-seq 12164  df-fac 12410  df-bc 12438  df-hash 12466  df-ram 14895
This theorem is referenced by:  ramcl  14930
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