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Theorem ramval 16173
 Description: The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 14-Sep-2020.)
Hypotheses
Ref Expression
ramval.c 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
ramval.t 𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}
Assertion
Ref Expression
ramval ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ*, < ))
Distinct variable groups:   𝑓,𝑐,𝑥,𝐶   𝑛,𝑐,𝑠,𝐹,𝑓,𝑥   𝑎,𝑏,𝑐,𝑓,𝑖,𝑛,𝑠,𝑥,𝑀   𝑅,𝑐,𝑓,𝑛,𝑠,𝑥   𝑉,𝑐,𝑓,𝑛,𝑠,𝑥
Allowed substitution hints:   𝐶(𝑖,𝑛,𝑠,𝑎,𝑏)   𝑅(𝑖,𝑎,𝑏)   𝑇(𝑥,𝑓,𝑖,𝑛,𝑠,𝑎,𝑏,𝑐)   𝐹(𝑖,𝑎,𝑏)   𝑉(𝑖,𝑎,𝑏)

Proof of Theorem ramval
Dummy variables 𝑦 𝑚 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ram 16166 . . 3 Ramsey = (𝑚 ∈ ℕ0, 𝑟 ∈ V ↦ inf({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))}, ℝ*, < ))
21a1i 11 . 2 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → Ramsey = (𝑚 ∈ ℕ0, 𝑟 ∈ V ↦ inf({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))}, ℝ*, < )))
3 simplrr 774 . . . . . . . . . . . 12 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝑟 = 𝐹)
43dmeqd 5660 . . . . . . . . . . 11 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝑟 = dom 𝐹)
5 simpll3 1207 . . . . . . . . . . . 12 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝐹:𝑅⟶ℕ0)
65fdmd 6391 . . . . . . . . . . 11 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝐹 = 𝑅)
74, 6eqtrd 2831 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝑟 = 𝑅)
8 simplrl 773 . . . . . . . . . . . . 13 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝑚 = 𝑀)
98eqeq2d 2805 . . . . . . . . . . . 12 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → ((♯‘𝑦) = 𝑚 ↔ (♯‘𝑦) = 𝑀))
109rabbidv 3425 . . . . . . . . . . 11 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚} = {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑀})
11 vex 3440 . . . . . . . . . . . 12 𝑠 ∈ V
12 simpll1 1205 . . . . . . . . . . . 12 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ ℕ0)
13 ramval.c . . . . . . . . . . . . 13 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
1413hashbcval 16167 . . . . . . . . . . . 12 ((𝑠 ∈ V ∧ 𝑀 ∈ ℕ0) → (𝑠𝐶𝑀) = {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑀})
1511, 12, 14sylancr 587 . . . . . . . . . . 11 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → (𝑠𝐶𝑀) = {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑀})
1610, 15eqtr4d 2834 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚} = (𝑠𝐶𝑀))
177, 16oveq12d 7034 . . . . . . . . 9 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚}) = (𝑅𝑚 (𝑠𝐶𝑀)))
1817raleqdv 3375 . . . . . . . 8 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → (∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)) ↔ ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐))))
19 simpr 485 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑟 = 𝐹) → 𝑟 = 𝐹)
2019dmeqd 5660 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑟 = 𝐹) → dom 𝑟 = dom 𝐹)
21 fdm 6390 . . . . . . . . . . . . 13 (𝐹:𝑅⟶ℕ0 → dom 𝐹 = 𝑅)
22213ad2ant3 1128 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → dom 𝐹 = 𝑅)
2320, 22sylan9eqr 2853 . . . . . . . . . . 11 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) → dom 𝑟 = 𝑅)
2423ad2antrr 722 . . . . . . . . . 10 (((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) → dom 𝑟 = 𝑅)
253ad2antrr 722 . . . . . . . . . . . . . 14 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑟 = 𝐹)
2625fveq1d 6540 . . . . . . . . . . . . 13 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑟𝑐) = (𝐹𝑐))
2726breq1d 4972 . . . . . . . . . . . 12 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → ((𝑟𝑐) ≤ (♯‘𝑥) ↔ (𝐹𝑐) ≤ (♯‘𝑥)))
288ad2antrr 722 . . . . . . . . . . . . . . . 16 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑚 = 𝑀)
2928oveq2d 7032 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑚) = (𝑥𝐶𝑀))
30 vex 3440 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
3112ad2antrr 722 . . . . . . . . . . . . . . . . 17 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑀 ∈ ℕ0)
3228, 31eqeltrd 2883 . . . . . . . . . . . . . . . 16 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑚 ∈ ℕ0)
3313hashbcval 16167 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ V ∧ 𝑚 ∈ ℕ0) → (𝑥𝐶𝑚) = {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚})
3430, 32, 33sylancr 587 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑚) = {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚})
3529, 34eqtr3d 2833 . . . . . . . . . . . . . 14 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑀) = {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚})
3635sseq1d 3919 . . . . . . . . . . . . 13 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → ((𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}) ↔ {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚} ⊆ (𝑓 “ {𝑐})))
37 rabss 3969 . . . . . . . . . . . . . 14 ({𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚} ⊆ (𝑓 “ {𝑐}) ↔ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚𝑦 ∈ (𝑓 “ {𝑐})))
3835eleq2d 2868 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑦 ∈ (𝑥𝐶𝑀) ↔ 𝑦 ∈ {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚}))
39 rabid 3337 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚} ↔ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚))
4038, 39syl6bb 288 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑦 ∈ (𝑥𝐶𝑀) ↔ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)))
4140biimpar 478 . . . . . . . . . . . . . . . . . . 19 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)) → 𝑦 ∈ (𝑥𝐶𝑀))
42 elpwi 4463 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ 𝒫 𝑠𝑥𝑠)
4342adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑥𝑠)
4413hashbcss 16169 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠 ∈ V ∧ 𝑥𝑠𝑀 ∈ ℕ0) → (𝑥𝐶𝑀) ⊆ (𝑠𝐶𝑀))
4511, 43, 31, 44mp3an2i 1458 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑀) ⊆ (𝑠𝐶𝑀))
4645sselda 3889 . . . . . . . . . . . . . . . . . . 19 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ 𝑦 ∈ (𝑥𝐶𝑀)) → 𝑦 ∈ (𝑠𝐶𝑀))
4741, 46syldan 591 . . . . . . . . . . . . . . . . . 18 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)) → 𝑦 ∈ (𝑠𝐶𝑀))
48 elmapi 8278 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀)) → 𝑓:(𝑠𝐶𝑀)⟶𝑅)
4948ad3antlr 727 . . . . . . . . . . . . . . . . . . 19 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)) → 𝑓:(𝑠𝐶𝑀)⟶𝑅)
50 ffn 6382 . . . . . . . . . . . . . . . . . . 19 (𝑓:(𝑠𝐶𝑀)⟶𝑅𝑓 Fn (𝑠𝐶𝑀))
51 fniniseg 6695 . . . . . . . . . . . . . . . . . . 19 (𝑓 Fn (𝑠𝐶𝑀) → (𝑦 ∈ (𝑓 “ {𝑐}) ↔ (𝑦 ∈ (𝑠𝐶𝑀) ∧ (𝑓𝑦) = 𝑐)))
5249, 50, 513syl 18 . . . . . . . . . . . . . . . . . 18 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)) → (𝑦 ∈ (𝑓 “ {𝑐}) ↔ (𝑦 ∈ (𝑠𝐶𝑀) ∧ (𝑓𝑦) = 𝑐)))
5347, 52mpbirand 703 . . . . . . . . . . . . . . . . 17 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)) → (𝑦 ∈ (𝑓 “ {𝑐}) ↔ (𝑓𝑦) = 𝑐))
5453anassrs 468 . . . . . . . . . . . . . . . 16 ((((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ 𝑦 ∈ 𝒫 𝑥) ∧ (♯‘𝑦) = 𝑚) → (𝑦 ∈ (𝑓 “ {𝑐}) ↔ (𝑓𝑦) = 𝑐))
5554pm5.74da 800 . . . . . . . . . . . . . . 15 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ 𝑦 ∈ 𝒫 𝑥) → (((♯‘𝑦) = 𝑚𝑦 ∈ (𝑓 “ {𝑐})) ↔ ((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))
5655ralbidva 3163 . . . . . . . . . . . . . 14 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚𝑦 ∈ (𝑓 “ {𝑐})) ↔ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))
5737, 56syl5bb 284 . . . . . . . . . . . . 13 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → ({𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚} ⊆ (𝑓 “ {𝑐}) ↔ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))
5836, 57bitr2d 281 . . . . . . . . . . . 12 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐) ↔ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))
5927, 58anbi12d 630 . . . . . . . . . . 11 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)) ↔ ((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
6059rexbidva 3259 . . . . . . . . . 10 (((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) → (∃𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)) ↔ ∃𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
6124, 60rexeqbidv 3362 . . . . . . . . 9 (((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) → (∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)) ↔ ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
6261ralbidva 3163 . . . . . . . 8 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → (∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)) ↔ ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
6318, 62bitrd 280 . . . . . . 7 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → (∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)) ↔ ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
6463imbi2d 342 . . . . . 6 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐))) ↔ (𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))))
6564albidv 1898 . . . . 5 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → (∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐))) ↔ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))))
6665rabbidva 3424 . . . 4 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) → {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))} = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))})
67 ramval.t . . . 4 𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}
6866, 67syl6eqr 2849 . . 3 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) → {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))} = 𝑇)
6968infeq1d 8787 . 2 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) → inf({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))}, ℝ*, < ) = inf(𝑇, ℝ*, < ))
70 simp1 1129 . 2 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝑀 ∈ ℕ0)
71 simp3 1131 . . 3 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝐹:𝑅⟶ℕ0)
72 simp2 1130 . . 3 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝑅𝑉)
73 fex 6855 . . 3 ((𝐹:𝑅⟶ℕ0𝑅𝑉) → 𝐹 ∈ V)
7471, 72, 73syl2anc 584 . 2 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝐹 ∈ V)
75 xrltso 12384 . . . 4 < Or ℝ*
7675infex 8803 . . 3 inf(𝑇, ℝ*, < ) ∈ V
7776a1i 11 . 2 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → inf(𝑇, ℝ*, < ) ∈ V)
782, 69, 70, 74, 77ovmpod 7158 1 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080  ∀wal 1520   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106  {crab 3109  Vcvv 3437   ⊆ wss 3859  𝒫 cpw 4453  {csn 4472   class class class wbr 4962  ◡ccnv 5442  dom cdm 5443   “ cima 5446   Fn wfn 6220  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016   ∈ cmpo 7018   ↑𝑚 cmap 8256  infcinf 8751  ℝ*cxr 10520   < clt 10521   ≤ cle 10522  ℕ0cn0 11745  ♯chash 13540   Ramsey cram 16164 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-pre-lttri 10457  ax-pre-lttrn 10458 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-po 5362  df-so 5363  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-sup 8752  df-inf 8753  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-ram 16166 This theorem is referenced by:  ramcl2lem  16174
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