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Theorem rami 15643
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 rami.g . . 3 (𝜑𝐺:(𝑆𝐶𝑀)⟶𝑅)
2 rami.r . . . 4 (𝜑𝑅𝑉)
3 ovex 6632 . . . 4 (𝑆𝐶𝑀) ∈ V
4 elmapg 7815 . . . 4 ((𝑅𝑉 ∧ (𝑆𝐶𝑀) ∈ V) → (𝐺 ∈ (𝑅𝑚 (𝑆𝐶𝑀)) ↔ 𝐺:(𝑆𝐶𝑀)⟶𝑅))
52, 3, 4sylancl 693 . . 3 (𝜑 → (𝐺 ∈ (𝑅𝑚 (𝑆𝐶𝑀)) ↔ 𝐺:(𝑆𝐶𝑀)⟶𝑅))
61, 5mpbird 247 . 2 (𝜑𝐺 ∈ (𝑅𝑚 (𝑆𝐶𝑀)))
7 rami.s . . 3 (𝜑𝑆𝑊)
8 rami.x . . . . 5 (𝜑 → (𝑀 Ramsey 𝐹) ∈ ℕ0)
9 rami.m . . . . . 6 (𝜑𝑀 ∈ ℕ0)
10 rami.f . . . . . 6 (𝜑𝐹:𝑅⟶ℕ0)
11 rami.c . . . . . . . 8 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
12 eqid 2621 . . . . . . . 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 15639 . . . . . . 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 15638 . . . . . . 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 271 . . . . . 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})))}))
169, 2, 10, 15syl3anc 1323 . . . . 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})))}))
178, 16mpbid 222 . . . 4 (𝜑 → (𝑀 Ramsey 𝐹) ∈ {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))})
18 breq1 4616 . . . . . . . 8 (𝑛 = (𝑀 Ramsey 𝐹) → (𝑛 ≤ (#‘𝑠) ↔ (𝑀 Ramsey 𝐹) ≤ (#‘𝑠)))
1918imbi1d 331 . . . . . . 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 1846 . . . . . 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 3346 . . . . 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 480 . . . 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 6148 . . . . . 6 (𝑠 = 𝑆 → (#‘𝑠) = (#‘𝑆))
2625breq2d 4625 . . . . 5 (𝑠 = 𝑆 → ((𝑀 Ramsey 𝐹) ≤ (#‘𝑠) ↔ (𝑀 Ramsey 𝐹) ≤ (#‘𝑆)))
27 oveq1 6611 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝐶𝑀) = (𝑆𝐶𝑀))
2827oveq2d 6620 . . . . . 6 (𝑠 = 𝑆 → (𝑅𝑚 (𝑠𝐶𝑀)) = (𝑅𝑚 (𝑆𝐶𝑀)))
29 pweq 4133 . . . . . . . 8 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
3029rexeqdv 3134 . . . . . . 7 (𝑠 = 𝑆 → (∃𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})) ↔ ∃𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
3130rexbidv 3045 . . . . . 6 (𝑠 = 𝑆 → (∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})) ↔ ∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
3228, 31raleqbidv 3141 . . . . 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 334 . . . 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 3279 . . 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})))))
357, 23, 24, 34syl3c 66 . 2 (𝜑 → ∀𝑓 ∈ (𝑅𝑚 (𝑆𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))
36 cnveq 5256 . . . . . . 7 (𝑓 = 𝐺𝑓 = 𝐺)
3736imaeq1d 5424 . . . . . 6 (𝑓 = 𝐺 → (𝑓 “ {𝑐}) = (𝐺 “ {𝑐}))
3837sseq2d 3612 . . . . 5 (𝑓 = 𝐺 → ((𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}) ↔ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐})))
3938anbi2d 739 . . . 4 (𝑓 = 𝐺 → (((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})) ↔ ((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐}))))
40392rexbidv 3050 . . 3 (𝑓 = 𝐺 → (∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})) ↔ ∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐}))))
4140rspcv 3291 . 2 (𝐺 ∈ (𝑅𝑚 (𝑆𝐶𝑀)) → (∀𝑓 ∈ (𝑅𝑚 (𝑆𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐}))))
426, 35, 41sylc 65 1 (𝜑 → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  c0 3891  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:  ramlb  15647  ramub1lem2  15655
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