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Theorem 0ramcl 13086
Description: Lemma for ramcl 13092: 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 5405 . . . . . . . 8  |-  ( F : R --> NN0  ->  F  Fn  R )
2 dffn4 5473 . . . . . . . 8  |-  ( F  Fn  R  <->  F : R -onto-> ran  F )
31, 2sylib 188 . . . . . . 7  |-  ( F : R --> NN0  ->  F : R -onto-> ran  F
)
43ad2antlr 707 . . . . . 6  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F : R -onto-> ran  F
)
5 foeq2 5464 . . . . . . 7  |-  ( R  =  (/)  ->  ( F : R -onto-> ran  F  <->  F : (/) -onto-> ran  F ) )
65adantl 452 . . . . . 6  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( F : R -onto-> ran  F  <->  F : (/) -onto-> ran  F
) )
74, 6mpbid 201 . . . . 5  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F : (/) -onto-> ran  F )
8 fo00 5525 . . . . . 6  |-  ( F : (/) -onto-> ran  F  <->  ( F  =  (/)  /\  ran  F  =  (/) ) )
98simplbi 446 . . . . 5  |-  ( F : (/) -onto-> ran  F  ->  F  =  (/) )
107, 9syl 15 . . . 4  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F  =  (/) )
1110oveq2d 5890 . . 3  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( 0 Ramsey  F )  =  ( 0 Ramsey  (/) ) )
12 0nn0 9996 . . . . 5  |-  0  e.  NN0
13 ram0 13085 . . . . 5  |-  ( 0  e.  NN0  ->  ( 0 Ramsey  (/) )  =  0 )
1412, 13ax-mp 8 . . . 4  |-  ( 0 Ramsey  (/) )  =  0
1514, 12eqeltri 2366 . . 3  |-  ( 0 Ramsey  (/) )  e.  NN0
1611, 15syl6eqel 2384 . 2  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( 0 Ramsey  F )  e.  NN0 )
17 0ram2 13084 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
18 frn 5411 . . . . . . 7  |-  ( F : R --> NN0  ->  ran 
F  C_  NN0 )
19183ad2ant3 978 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  NN0 )
20 nn0ssz 10060 . . . . . . . 8  |-  NN0  C_  ZZ
2119, 20syl6ss 3204 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  ZZ )
22 fdm 5409 . . . . . . . . . 10  |-  ( F : R --> NN0  ->  dom 
F  =  R )
23223ad2ant3 978 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =  R )
24 simp2 956 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  =/=  (/) )
2523, 24eqnetrd 2477 . . . . . . . 8  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =/=  (/) )
26 dm0rn0 4911 . . . . . . . . 9  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
2726necon3bii 2491 . . . . . . . 8  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
2825, 27sylib 188 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  =/=  (/) )
29 nn0ssre 9985 . . . . . . . . . 10  |-  NN0  C_  RR
3019, 29syl6ss 3204 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  RR )
31 simp1 955 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  e.  Fin )
3233ad2ant3 978 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  F : R -onto-> ran  F )
33 fofi 7158 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  F : R -onto-> ran  F
)  ->  ran  F  e. 
Fin )
3431, 32, 33syl2anc 642 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  e.  Fin )
35 fimaxre 9717 . . . . . . . . 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 1182 . . . . . . . 8  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ran  F A. y  e.  ran  F  y  <_  x )
37 ssrexv 3251 . . . . . . . 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 56 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ZZ  A. y  e. 
ran  F  y  <_  x )
39 suprzcl2 10324 . . . . . . 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 1182 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
4119, 40sseldd 3194 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  sup ( ran  F ,  RR ,  <  )  e.  NN0 )
4217, 41eqeltrd 2370 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  e.  NN0 )
43423expa 1151 . . 3  |-  ( ( ( R  e.  Fin  /\  R  =/=  (/) )  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )
4443an32s 779 . 2  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =/=  (/) )  -> 
( 0 Ramsey  F )  e.  NN0 )
4516, 44pm2.61dane 2537 1  |-  ( ( R  e.  Fin  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   class class class wbr 4039   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   -onto->wfo 5269  (class class class)co 5874   Fincfn 6879   supcsup 7209   RRcr 8752   0cc0 8753    < clt 8883    <_ cle 8884   NN0cn0 9981   ZZcz 10040   Ramsey cram 13062
This theorem is referenced by:  ramcl  13092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-fac 11305  df-bc 11332  df-hash 11354  df-ram 13064
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