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Theorem 0ramcl 14099
Description: Lemma for ramcl 14105: 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 5574 . . . . . . . 8  |-  ( F : R --> NN0  ->  F  Fn  R )
2 dffn4 5641 . . . . . . . 8  |-  ( F  Fn  R  <->  F : R -onto-> ran  F )
31, 2sylib 196 . . . . . . 7  |-  ( F : R --> NN0  ->  F : R -onto-> ran  F
)
43ad2antlr 726 . . . . . 6  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F : R -onto-> ran  F
)
5 foeq2 5632 . . . . . . 7  |-  ( R  =  (/)  ->  ( F : R -onto-> ran  F  <->  F : (/) -onto-> ran  F ) )
65adantl 466 . . . . . 6  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( F : R -onto-> ran  F  <->  F : (/) -onto-> ran  F
) )
74, 6mpbid 210 . . . . 5  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F : (/) -onto-> ran  F )
8 fo00 5689 . . . . . 6  |-  ( F : (/) -onto-> ran  F  <->  ( F  =  (/)  /\  ran  F  =  (/) ) )
98simplbi 460 . . . . 5  |-  ( F : (/) -onto-> ran  F  ->  F  =  (/) )
107, 9syl 16 . . . 4  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F  =  (/) )
1110oveq2d 6122 . . 3  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( 0 Ramsey  F )  =  ( 0 Ramsey  (/) ) )
12 0nn0 10609 . . . . 5  |-  0  e.  NN0
13 ram0 14098 . . . . 5  |-  ( 0  e.  NN0  ->  ( 0 Ramsey  (/) )  =  0 )
1412, 13ax-mp 5 . . . 4  |-  ( 0 Ramsey  (/) )  =  0
1514, 12eqeltri 2513 . . 3  |-  ( 0 Ramsey  (/) )  e.  NN0
1611, 15syl6eqel 2531 . 2  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( 0 Ramsey  F )  e.  NN0 )
17 0ram2 14097 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
18 frn 5580 . . . . . . 7  |-  ( F : R --> NN0  ->  ran 
F  C_  NN0 )
19183ad2ant3 1011 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  NN0 )
20 nn0ssz 10682 . . . . . . . 8  |-  NN0  C_  ZZ
2119, 20syl6ss 3383 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  ZZ )
22 fdm 5578 . . . . . . . . . 10  |-  ( F : R --> NN0  ->  dom 
F  =  R )
23223ad2ant3 1011 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =  R )
24 simp2 989 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  =/=  (/) )
2523, 24eqnetrd 2641 . . . . . . . 8  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =/=  (/) )
26 dm0rn0 5071 . . . . . . . . 9  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
2726necon3bii 2655 . . . . . . . 8  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
2825, 27sylib 196 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  =/=  (/) )
29 nn0ssre 10598 . . . . . . . . . 10  |-  NN0  C_  RR
3019, 29syl6ss 3383 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  RR )
31 simp1 988 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  e.  Fin )
3233ad2ant3 1011 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  F : R -onto-> ran  F )
33 fofi 7612 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  F : R -onto-> ran  F
)  ->  ran  F  e. 
Fin )
3431, 32, 33syl2anc 661 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  e.  Fin )
35 fimaxre 10292 . . . . . . . . 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 1218 . . . . . . . 8  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ran  F A. y  e.  ran  F  y  <_  x )
37 ssrexv 3432 . . . . . . . 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 60 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ZZ  A. y  e. 
ran  F  y  <_  x )
39 suprzcl2 10960 . . . . . . 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 1218 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
4119, 40sseldd 3372 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  sup ( ran  F ,  RR ,  <  )  e.  NN0 )
4217, 41eqeltrd 2517 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  e.  NN0 )
43423expa 1187 . . 3  |-  ( ( ( R  e.  Fin  /\  R  =/=  (/) )  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )
4443an32s 802 . 2  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =/=  (/) )  -> 
( 0 Ramsey  F )  e.  NN0 )
4516, 44pm2.61dane 2704 1  |-  ( ( R  e.  Fin  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2730   E.wrex 2731    C_ wss 3343   (/)c0 3652   class class class wbr 4307   dom cdm 4855   ran crn 4856    Fn wfn 5428   -->wf 5429   -onto->wfo 5431  (class class class)co 6106   Fincfn 7325   supcsup 7705   RRcr 9296   0cc0 9297    < clt 9433    <_ cle 9434   NN0cn0 10594   ZZcz 10661   Ramsey cram 14075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-2o 6936  df-oadd 6939  df-er 7116  df-map 7231  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-sup 7706  df-card 8124  df-cda 8352  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-n0 10595  df-z 10662  df-uz 10877  df-rp 11007  df-fz 11453  df-seq 11822  df-fac 12067  df-bc 12094  df-hash 12119  df-ram 14077
This theorem is referenced by:  ramcl  14105
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