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Theorem ramub 15641
Description: The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 22-Apr-2015.)
Hypotheses
Ref Expression
rami.c 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
rami.m (𝜑𝑀 ∈ ℕ0)
rami.r (𝜑𝑅𝑉)
rami.f (𝜑𝐹:𝑅⟶ℕ0)
ramub.n (𝜑𝑁 ∈ ℕ0)
ramub.i ((𝜑 ∧ (𝑁 ≤ (#‘𝑠) ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))
Assertion
Ref Expression
ramub (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁)
Distinct variable groups:   𝑓,𝑐,𝑠,𝑥,𝐶   𝜑,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:   𝜑(𝑖,𝑎,𝑏)   𝐶(𝑖,𝑎,𝑏)   𝑅(𝑖,𝑎,𝑏)   𝐹(𝑖,𝑎,𝑏)   𝑁(𝑏)   𝑉(𝑖,𝑎,𝑏)

Proof of Theorem ramub
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 rami.m . 2 (𝜑𝑀 ∈ ℕ0)
2 rami.r . 2 (𝜑𝑅𝑉)
3 rami.f . 2 (𝜑𝐹:𝑅⟶ℕ0)
4 ramub.n . . 3 (𝜑𝑁 ∈ ℕ0)
5 elmapi 7823 . . . . . 6 (𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀)) → 𝑓:(𝑠𝐶𝑀)⟶𝑅)
6 ramub.i . . . . . . . 8 ((𝜑 ∧ (𝑁 ≤ (#‘𝑠) ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))
76ancom2s 843 . . . . . . 7 ((𝜑 ∧ (𝑓:(𝑠𝐶𝑀)⟶𝑅𝑁 ≤ (#‘𝑠))) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))
87expr 642 . . . . . 6 ((𝜑𝑓:(𝑠𝐶𝑀)⟶𝑅) → (𝑁 ≤ (#‘𝑠) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
95, 8sylan2 491 . . . . 5 ((𝜑𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) → (𝑁 ≤ (#‘𝑠) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
109ralrimdva 2963 . . . 4 (𝜑 → (𝑁 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
1110alrimiv 1852 . . 3 (𝜑 → ∀𝑠(𝑁 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
12 breq1 4616 . . . . . 6 (𝑛 = 𝑁 → (𝑛 ≤ (#‘𝑠) ↔ 𝑁 ≤ (#‘𝑠)))
1312imbi1d 331 . . . . 5 (𝑛 = 𝑁 → ((𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))) ↔ (𝑁 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))))
1413albidv 1846 . . . 4 (𝑛 = 𝑁 → (∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))) ↔ ∀𝑠(𝑁 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))))
1514elrab 3346 . . 3 (𝑁 ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))} ↔ (𝑁 ∈ ℕ0 ∧ ∀𝑠(𝑁 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))))
164, 11, 15sylanbrc 697 . 2 (𝜑𝑁 ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))})
17 rami.c . . 3 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
18 eqid 2621 . . 3 {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))} = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}
1917, 18ramtub 15640 . 2 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ 𝑁 ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}) → (𝑀 Ramsey 𝐹) ≤ 𝑁)
201, 2, 3, 16, 19syl31anc 1326 1 (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130  {csn 4148   class class class wbr 4613  ccnv 5073  cima 5077  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802  cle 10019  0cn0 11236  #chash 13057   Ramsey cram 15627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-ram 15629
This theorem is referenced by:  ramub2  15642  0ram  15648  ram0  15650  ramz2  15652
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