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Theorem ramval 15631
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 15624 . . 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 800 . . . . . . . . . . . 12 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝑟 = 𝐹)
43dmeqd 5291 . . . . . . . . . . 11 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝑟 = dom 𝐹)
5 simpll3 1100 . . . . . . . . . . . 12 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝐹:𝑅⟶ℕ0)
6 fdm 6010 . . . . . . . . . . . 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 2660 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝑟 = 𝑅)
9 simplrl 799 . . . . . . . . . . . . 13 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝑚 = 𝑀)
109eqeq2d 2636 . . . . . . . . . . . 12 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → ((#‘𝑦) = 𝑚 ↔ (#‘𝑦) = 𝑀))
1110rabbidv 3182 . . . . . . . . . . 11 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑚} = {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑀})
12 vex 3194 . . . . . . . . . . . 12 𝑠 ∈ V
13 simpll1 1098 . . . . . . . . . . . 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 15625 . . . . . . . . . . . 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 2663 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑚} = (𝑠𝐶𝑀))
188, 17oveq12d 6623 . . . . . . . . 9 ((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑚}) = (𝑅𝑚 (𝑠𝐶𝑀)))
1918raleqdv 3138 . . . . . . . 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 477 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑟 = 𝐹) → 𝑟 = 𝐹)
2120dmeqd 5291 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑟 = 𝐹) → dom 𝑟 = dom 𝐹)
2263ad2ant3 1082 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → dom 𝐹 = 𝑅)
2321, 22sylan9eqr 2682 . . . . . . . . . . 11 (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) → dom 𝑟 = 𝑅)
2423ad2antrr 761 . . . . . . . . . 10 (((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) → dom 𝑟 = 𝑅)
253ad2antrr 761 . . . . . . . . . . . . . 14 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑟 = 𝐹)
2625fveq1d 6152 . . . . . . . . . . . . 13 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑟𝑐) = (𝐹𝑐))
2726breq1d 4628 . . . . . . . . . . . 12 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → ((𝑟𝑐) ≤ (#‘𝑥) ↔ (𝐹𝑐) ≤ (#‘𝑥)))
289ad2antrr 761 . . . . . . . . . . . . . . . 16 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑚 = 𝑀)
2928oveq2d 6621 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑚) = (𝑥𝐶𝑀))
30 vex 3194 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
3113ad2antrr 761 . . . . . . . . . . . . . . . . 17 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑀 ∈ ℕ0)
3228, 31eqeltrd 2704 . . . . . . . . . . . . . . . 16 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑚 ∈ ℕ0)
3314hashbcval 15625 . . . . . . . . . . . . . . . 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 2662 . . . . . . . . . . . . . 14 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑀) = {𝑦 ∈ 𝒫 𝑥 ∣ (#‘𝑦) = 𝑚})
3635sseq1d 3616 . . . . . . . . . . . . 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 3663 . . . . . . . . . . . . . 14 ({𝑦 ∈ 𝒫 𝑥 ∣ (#‘𝑦) = 𝑚} ⊆ (𝑓 “ {𝑐}) ↔ ∀𝑦 ∈ 𝒫 𝑥((#‘𝑦) = 𝑚𝑦 ∈ (𝑓 “ {𝑐})))
38 elmapi 7824 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀)) → 𝑓:(𝑠𝐶𝑀)⟶𝑅)
3938ad3antlr 766 . . . . . . . . . . . . . . . . . . 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 6004 . . . . . . . . . . . . . . . . . . 19 (𝑓:(𝑠𝐶𝑀)⟶𝑅𝑓 Fn (𝑠𝐶𝑀))
41 fniniseg 6295 . . . . . . . . . . . . . . . . . . 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 2689 . . . . . . . . . . . . . . . . . . . . . 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 3111 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ {𝑦 ∈ 𝒫 𝑥 ∣ (#‘𝑦) = 𝑚} ↔ (𝑦 ∈ 𝒫 𝑥 ∧ (#‘𝑦) = 𝑚))
4543, 44syl6bb 276 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑦 ∈ (𝑥𝐶𝑀) ↔ (𝑦 ∈ 𝒫 𝑥 ∧ (#‘𝑦) = 𝑚)))
4645biimpar 502 . . . . . . . . . . . . . . . . . . . 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 4145 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ 𝒫 𝑠𝑥𝑠)
4948adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑥𝑠)
5014hashbcss 15627 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ V ∧ 𝑥𝑠𝑀 ∈ ℕ0) → (𝑥𝐶𝑀) ⊆ (𝑠𝐶𝑀))
5147, 49, 31, 50syl3anc 1323 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑀) ⊆ (𝑠𝐶𝑀))
5251sselda 3588 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ 𝑦 ∈ (𝑥𝐶𝑀)) → 𝑦 ∈ (𝑠𝐶𝑀))
5346, 52syldan 487 . . . . . . . . . . . . . . . . . . 19 (((((((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (#‘𝑦) = 𝑚)) → 𝑦 ∈ (𝑠𝐶𝑀))
5453biantrurd 529 . . . . . . . . . . . . . . . . . 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 271 . . . . . . . . . . . . . . . . 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 679 . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . 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 2984 . . . . . . . . . . . . . 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 272 . . . . . . . . . . . . 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 269 . . . . . . . . . . . 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 746 . . . . . . . . . . 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 3047 . . . . . . . . . 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 3147 . . . . . . . . 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 2984 . . . . . . . 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 268 . . . . . . 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 330 . . . . . 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 1851 . . . . 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 3181 . . . 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 2678 . . 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 8328 . 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 1059 . 2 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝑀 ∈ ℕ0)
73 simp3 1061 . . 3 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝐹:𝑅⟶ℕ0)
74 simp2 1060 . . 3 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝑅𝑉)
75 fex 6445 . . 3 ((𝐹:𝑅⟶ℕ0𝑅𝑉) → 𝐹 ∈ V)
7673, 74, 75syl2anc 692 . 2 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → 𝐹 ∈ V)
77 xrltso 11918 . . . 4 < Or ℝ*
7877infex 8344 . . 3 inf(𝑇, ℝ*, < ) ∈ V
7978a1i 11 . 2 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → inf(𝑇, ℝ*, < ) ∈ V)
802, 71, 72, 76, 79ovmpt2d 6742 1 ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1992  wral 2912  wrex 2913  {crab 2916  Vcvv 3191  wss 3560  𝒫 cpw 4135  {csn 4153   class class class wbr 4618  ccnv 5078  dom cdm 5079  cima 5082   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  cmpt2 6607  𝑚 cmap 7803  infcinf 8292  *cxr 10018   < clt 10019  cle 10020  0cn0 11237  #chash 13054   Ramsey cram 15622
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-pre-lttri 9955  ax-pre-lttrn 9956
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-sup 8293  df-inf 8294  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-ram 15624
This theorem is referenced by:  ramcl2lem  15632
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