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Theorem ramval 15550
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 15543 . . 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 797 . . . . . . . . . . . 12 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝑟 = 𝐹)
43dmeqd 5248 . . . . . . . . . . 11 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝑟 = dom 𝐹)
5 simpll3 1095 . . . . . . . . . . . 12 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝐹:𝑅⟶ℕ0)
6 fdm 5964 . . . . . . . . . . . 12 (𝐹:𝑅⟶ℕ0 → dom 𝐹 = 𝑅)
75, 6syl 17 . . . . . . . . . . 11 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝐹 = 𝑅)
84, 7eqtrd 2644 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝑟 = 𝑅)
9 simplrl 796 . . . . . . . . . . . . 13 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝑚 = 𝑀)
109eqeq2d 2620 . . . . . . . . . . . 12 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → ((#‘𝑦) = 𝑚 ↔ (#‘𝑦) = 𝑀))
1110rabbidv 3164 . . . . . . . . . . 11 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑚} = {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑀})
12 vex 3176 . . . . . . . . . . . 12 𝑠 ∈ V
13 simpll1 1093 . . . . . . . . . . . 12 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ ℕ0)
14 ramval.c . . . . . . . . . . . . 13 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
1514hashbcval 15544 . . . . . . . . . . . 12 ((𝑠 ∈ V ∧ 𝑀 ∈ ℕ0) → (𝑠𝐶𝑀) = {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑀})
1612, 13, 15sylancr 694 . . . . . . . . . . 11 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → (𝑠𝐶𝑀) = {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑀})
1711, 16eqtr4d 2647 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑚} = (𝑠𝐶𝑀))
188, 17oveq12d 6567 . . . . . . . . 9 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑚}) = (𝑅𝑚 (𝑠𝐶𝑀)))
1918raleqdv 3121 . . . . . . . 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))))
20 simpr 476 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑟 = 𝐹) → 𝑟 = 𝐹)
2120dmeqd 5248 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑟 = 𝐹) → dom 𝑟 = dom 𝐹)
2263ad2ant3 1077 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → dom 𝐹 = 𝑅)
2321, 22sylan9eqr 2666 . . . . . . . . . . 11 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) → dom 𝑟 = 𝑅)
2423ad2antrr 758 . . . . . . . . . 10 (((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) → dom 𝑟 = 𝑅)
253ad2antrr 758 . . . . . . . . . . . . . 14 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑟 = 𝐹)
2625fveq1d 6105 . . . . . . . . . . . . 13 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑟𝑐) = (𝐹𝑐))
2726breq1d 4593 . . . . . . . . . . . 12 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → ((𝑟𝑐) ≤ (#‘𝑥) ↔ (𝐹𝑐) ≤ (#‘𝑥)))
289ad2antrr 758 . . . . . . . . . . . . . . . 16 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑚 = 𝑀)
2928oveq2d 6565 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑚) = (𝑥𝐶𝑀))
30 vex 3176 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
3113ad2antrr 758 . . . . . . . . . . . . . . . . 17 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑀 ∈ ℕ0)
3228, 31eqeltrd 2688 . . . . . . . . . . . . . . . 16 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑚 ∈ ℕ0)
3314hashbcval 15544 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ V ∧ 𝑚 ∈ ℕ0) → (𝑥𝐶𝑚) = {𝑦 ∈ 𝒫 𝑥 ∣ (#‘𝑦) = 𝑚})
3430, 32, 33sylancr 694 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑚) = {𝑦 ∈ 𝒫 𝑥 ∣ (#‘𝑦) = 𝑚})
3529, 34eqtr3d 2646 . . . . . . . . . . . . . 14 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑀) = {𝑦 ∈ 𝒫 𝑥 ∣ (#‘𝑦) = 𝑚})
3635sseq1d 3595 . . . . . . . . . . . . 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 3642 . . . . . . . . . . . . . 14 ({𝑦 ∈ 𝒫 𝑥 ∣ (#‘𝑦) = 𝑚} ⊆ (𝑓 “ {𝑐}) ↔ ∀𝑦 ∈ 𝒫 𝑥((#‘𝑦) = 𝑚𝑦 ∈ (𝑓 “ {𝑐})))
38 elmapi 7765 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀)) → 𝑓:(𝑠𝐶𝑀)⟶𝑅)
3938ad3antlr 763 . . . . . . . . . . . . . . . . . . 19 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (#‘𝑦) = 𝑚)) → 𝑓:(𝑠𝐶𝑀)⟶𝑅)
40 ffn 5958 . . . . . . . . . . . . . . . . . . 19 (𝑓:(𝑠𝐶𝑀)⟶𝑅𝑓 Fn (𝑠𝐶𝑀))
41 fniniseg 6246 . . . . . . . . . . . . . . . . . . 19 (𝑓 Fn (𝑠𝐶𝑀) → (𝑦 ∈ (𝑓 “ {𝑐}) ↔ (𝑦 ∈ (𝑠𝐶𝑀) ∧ (𝑓𝑦) = 𝑐)))
4239, 40, 413syl 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)))
4335eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑦 ∈ (𝑥𝐶𝑀) ↔ 𝑦 ∈ {𝑦 ∈ 𝒫 𝑥 ∣ (#‘𝑦) = 𝑚}))
44 rabid 3095 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ {𝑦 ∈ 𝒫 𝑥 ∣ (#‘𝑦) = 𝑚} ↔ (𝑦 ∈ 𝒫 𝑥 ∧ (#‘𝑦) = 𝑚))
4543, 44syl6bb 275 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑦 ∈ (𝑥𝐶𝑀) ↔ (𝑦 ∈ 𝒫 𝑥 ∧ (#‘𝑦) = 𝑚)))
4645biimpar 501 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (#‘𝑦) = 𝑚)) → 𝑦 ∈ (𝑥𝐶𝑀))
4712a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑠 ∈ V)
48 elpwi 4117 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ 𝒫 𝑠𝑥𝑠)
4948adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑥𝑠)
5014hashbcss 15546 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ V ∧ 𝑥𝑠𝑀 ∈ ℕ0) → (𝑥𝐶𝑀) ⊆ (𝑠𝐶𝑀))
5147, 49, 31, 50syl3anc 1318 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑀) ⊆ (𝑠𝐶𝑀))
5251sselda 3568 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ 𝑦 ∈ (𝑥𝐶𝑀)) → 𝑦 ∈ (𝑠𝐶𝑀))
5346, 52syldan 486 . . . . . . . . . . . . . . . . . . 19 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (#‘𝑦) = 𝑚)) → 𝑦 ∈ (𝑠𝐶𝑀))
5453biantrurd 528 . . . . . . . . . . . . . . . . . 18 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (#‘𝑦) = 𝑚)) → ((𝑓𝑦) = 𝑐 ↔ (𝑦 ∈ (𝑠𝐶𝑀) ∧ (𝑓𝑦) = 𝑐)))
5542, 54bitr4d 270 . . . . . . . . . . . . . . . . 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))
5655anassrs 678 . . . . . . . . . . . . . . . 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))
5756pm5.74da 719 . . . . . . . . . . . . . . 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)))
5857ralbidva 2968 . . . . . . . . . . . . . 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)))
5937, 58syl5bb 271 . . . . . . . . . . . . 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)))
6036, 59bitr2d 268 . . . . . . . . . . . 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})))
6127, 60anbi12d 743 . . . . . . . . . . 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}))))
6261rexbidva 3031 . . . . . . . . . 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}))))
6324, 62rexeqbidv 3130 . . . . . . . . 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}))))
6463ralbidva 2968 . . . . . . . 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}))))
6519, 64bitrd 267 . . . . . . 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}))))
6665imbi2d 329 . . . . . 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})))))
6766albidv 1836 . . . . 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})))))
6867rabbidva 3163 . . . 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})))})
69 ramval.t . . . 4 𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}
7068, 69syl6eqr 2662 . . 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)
7170infeq1d 8266 . 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*, < ))
72 simp1 1054 . 2 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝑀 ∈ ℕ0)
73 simp3 1056 . . 3 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝐹:𝑅⟶ℕ0)
74 simp2 1055 . . 3 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝑅𝑉)
75 fex 6394 . . 3 ((𝐹:𝑅⟶ℕ0𝑅𝑉) → 𝐹 ∈ V)
7673, 74, 75syl2anc 691 . 2 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝐹 ∈ V)
77 xrltso 11850 . . . 4 < Or ℝ*
7877infex 8282 . . 3 inf(𝑇, ℝ*, < ) ∈ V
7978a1i 11 . 2 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → inf(𝑇, ℝ*, < ) ∈ V)
802, 71, 72, 76, 79ovmpt2d 6686 1 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031  wal 1473   = wceq 1475  wcel 1977  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  wss 3540  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  ccnv 5037  dom cdm 5038  cima 5041   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  cmpt2 6551  𝑚 cmap 7744  infcinf 8230  *cxr 9952   < clt 9953  cle 9954  0cn0 11169  #chash 12979   Ramsey cram 15541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-ram 15543
This theorem is referenced by:  ramcl2lem  15551
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