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Theorem rami 16328
Description: The defining property of a Ramsey number. (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)
rami.x (𝜑 → (𝑀 Ramsey 𝐹) ∈ ℕ0)
rami.s (𝜑𝑆𝑊)
rami.l (𝜑 → (𝑀 Ramsey 𝐹) ≤ (♯‘𝑆))
rami.g (𝜑𝐺:(𝑆𝐶𝑀)⟶𝑅)
Assertion
Ref Expression
rami (𝜑 → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐})))
Distinct variable groups:   𝑥,𝑐,𝐶   𝐺,𝑐,𝑥   𝜑,𝑐,𝑥   𝑆,𝑐,𝑥   𝐹,𝑐,𝑥   𝑎,𝑏,𝑐,𝑖,𝑥,𝑀   𝑅,𝑐,𝑥   𝑉,𝑐,𝑥
Allowed substitution hints:   𝜑(𝑖,𝑎,𝑏)   𝐶(𝑖,𝑎,𝑏)   𝑅(𝑖,𝑎,𝑏)   𝑆(𝑖,𝑎,𝑏)   𝐹(𝑖,𝑎,𝑏)   𝐺(𝑖,𝑎,𝑏)   𝑉(𝑖,𝑎,𝑏)   𝑊(𝑥,𝑖,𝑎,𝑏,𝑐)

Proof of Theorem rami
Dummy variables 𝑓 𝑛 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnveq 5717 . . . . . 6 (𝑓 = 𝐺𝑓 = 𝐺)
21imaeq1d 5901 . . . . 5 (𝑓 = 𝐺 → (𝑓 “ {𝑐}) = (𝐺 “ {𝑐}))
32sseq2d 3975 . . . 4 (𝑓 = 𝐺 → ((𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}) ↔ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐})))
43anbi2d 631 . . 3 (𝑓 = 𝐺 → (((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})) ↔ ((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐}))))
542rexbidv 3286 . 2 (𝑓 = 𝐺 → (∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})) ↔ ∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐}))))
6 rami.s . . 3 (𝜑𝑆𝑊)
7 rami.x . . . . 5 (𝜑 → (𝑀 Ramsey 𝐹) ∈ ℕ0)
8 rami.m . . . . . 6 (𝜑𝑀 ∈ ℕ0)
9 rami.r . . . . . 6 (𝜑𝑅𝑉)
10 rami.f . . . . . 6 (𝜑𝐹:𝑅⟶ℕ0)
11 rami.c . . . . . . . 8 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
12 eqid 2821 . . . . . . . 8 {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})))}
1311, 12ramtcl2 16324 . . . . . . 7 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ ℕ0 ↔ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))} ≠ ∅))
1411, 12ramtcl 16323 . . . . . . 7 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ {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})))} ≠ ∅))
1513, 14bitr4d 285 . . . . . 6 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ ℕ0 ↔ (𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}))
168, 9, 10, 15syl3anc 1368 . . . . 5 (𝜑 → ((𝑀 Ramsey 𝐹) ∈ ℕ0 ↔ (𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}))
177, 16mpbid 235 . . . 4 (𝜑 → (𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))})
18 breq1 5042 . . . . . . . 8 (𝑛 = (𝑀 Ramsey 𝐹) → (𝑛 ≤ (♯‘𝑠) ↔ (𝑀 Ramsey 𝐹) ≤ (♯‘𝑠)))
1918imbi1d 345 . . . . . . 7 (𝑛 = (𝑀 Ramsey 𝐹) → ((𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))) ↔ ((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))))
2019albidv 1922 . . . . . 6 (𝑛 = (𝑀 Ramsey 𝐹) → (∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))) ↔ ∀𝑠((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))))
2120elrab 3657 . . . . 5 ((𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))} ↔ ((𝑀 Ramsey 𝐹) ∈ ℕ0 ∧ ∀𝑠((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))))
2221simprbi 500 . . . 4 ((𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))} → ∀𝑠((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
2317, 22syl 17 . . 3 (𝜑 → ∀𝑠((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
24 rami.l . . 3 (𝜑 → (𝑀 Ramsey 𝐹) ≤ (♯‘𝑆))
25 fveq2 6643 . . . . . 6 (𝑠 = 𝑆 → (♯‘𝑠) = (♯‘𝑆))
2625breq2d 5051 . . . . 5 (𝑠 = 𝑆 → ((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) ↔ (𝑀 Ramsey 𝐹) ≤ (♯‘𝑆)))
27 oveq1 7137 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝐶𝑀) = (𝑆𝐶𝑀))
2827oveq2d 7146 . . . . . 6 (𝑠 = 𝑆 → (𝑅m (𝑠𝐶𝑀)) = (𝑅m (𝑆𝐶𝑀)))
29 pweq 4528 . . . . . . . 8 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
3029rexeqdv 3397 . . . . . . 7 (𝑠 = 𝑆 → (∃𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})) ↔ ∃𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
3130rexbidv 3283 . . . . . 6 (𝑠 = 𝑆 → (∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})) ↔ ∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
3228, 31raleqbidv 3386 . . . . 5 (𝑠 = 𝑆 → (∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})) ↔ ∀𝑓 ∈ (𝑅m (𝑆𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
3326, 32imbi12d 348 . . . 4 (𝑠 = 𝑆 → (((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))) ↔ ((𝑀 Ramsey 𝐹) ≤ (♯‘𝑆) → ∀𝑓 ∈ (𝑅m (𝑆𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))))
3433spcgv 3572 . . 3 (𝑆𝑊 → (∀𝑠((𝑀 Ramsey 𝐹) ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))) → ((𝑀 Ramsey 𝐹) ≤ (♯‘𝑆) → ∀𝑓 ∈ (𝑅m (𝑆𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))))
356, 23, 24, 34syl3c 66 . 2 (𝜑 → ∀𝑓 ∈ (𝑅m (𝑆𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))
36 rami.g . . 3 (𝜑𝐺:(𝑆𝐶𝑀)⟶𝑅)
37 ovex 7163 . . . 4 (𝑆𝐶𝑀) ∈ V
38 elmapg 8394 . . . 4 ((𝑅𝑉 ∧ (𝑆𝐶𝑀) ∈ V) → (𝐺 ∈ (𝑅m (𝑆𝐶𝑀)) ↔ 𝐺:(𝑆𝐶𝑀)⟶𝑅))
399, 37, 38sylancl 589 . . 3 (𝜑 → (𝐺 ∈ (𝑅m (𝑆𝐶𝑀)) ↔ 𝐺:(𝑆𝐶𝑀)⟶𝑅))
4036, 39mpbird 260 . 2 (𝜑𝐺 ∈ (𝑅m (𝑆𝐶𝑀)))
415, 35, 40rspcdva 3602 1 (𝜑 → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wcel 2115  wne 3007  wral 3126  wrex 3127  {crab 3130  Vcvv 3471  wss 3910  c0 4266  𝒫 cpw 4512  {csn 4540   class class class wbr 5039  ccnv 5527  cima 5531  wf 6324  cfv 6328  (class class class)co 7130  cmpo 7132  m cmap 8381  cle 10653  0cn0 11875  chash 13674   Ramsey cram 16312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-map 8383  df-en 8485  df-dom 8486  df-sdom 8487  df-sup 8882  df-inf 8883  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-n0 11876  df-z 11960  df-uz 12222  df-ram 16314
This theorem is referenced by:  ramlb  16332  ramub1lem2  16340
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