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Theorem ramval 16948

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 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐})))}

**Assertion**
Ref	Expression
ramval	⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹: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.

Step	Hyp	Ref	Expression
1		df-ram 16941	. . 3 ⊢ Ramsey = (𝑚 ∈ ℕ₀, 𝑟 ∈ V ↦ inf({𝑛 ∈ ℕ₀ ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟 ↑_m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)))}, ℝ^*, < ))
2	1	a1i 11	. 2 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) → Ramsey = (𝑚 ∈ ℕ₀, 𝑟 ∈ V ↦ inf({𝑛 ∈ ℕ₀ ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟 ↑_m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)))}, ℝ^*, < )))
3		simplrr 775	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → 𝑟 = 𝐹)
4	3	dmeqd 5898	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → dom 𝑟 = dom 𝐹)
5		simpll3 1211	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → 𝐹:𝑅⟶ℕ₀)
6	5	fdmd 6721	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → dom 𝐹 = 𝑅)
7	4, 6	eqtrd 2766	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → dom 𝑟 = 𝑅)
8		simplrl 774	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → 𝑚 = 𝑀)
9	8	eqeq2d 2737	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → ((♯‘𝑦) = 𝑚 ↔ (♯‘𝑦) = 𝑀))
10	9	rabbidv 3434	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚} = {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑀})
11		vex 3472	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
12		simpll1 1209	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → 𝑀 ∈ ℕ₀)
13		ramval.c	. . . . . . . . . . . . 13 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
14	13	hashbcval 16942	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ V ∧ 𝑀 ∈ ℕ₀) → (𝑠𝐶𝑀) = {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑀})
15	11, 12, 14	sylancr 586	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → (𝑠𝐶𝑀) = {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑀})
16	10, 15	eqtr4d 2769	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚} = (𝑠𝐶𝑀))
17	7, 16	oveq12d 7422	. . . . . . . . 9 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → (dom 𝑟 ↑_m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚}) = (𝑅 ↑_m (𝑠𝐶𝑀)))
18	17	raleqdv 3319	. . . . . . . 8 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → (∀𝑓 ∈ (dom 𝑟 ↑_m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)) ↔ ∀𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐))))
19		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑟 = 𝐹) → 𝑟 = 𝐹)
20	19	dmeqd 5898	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑟 = 𝐹) → dom 𝑟 = dom 𝐹)
21		fdm 6719	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑅⟶ℕ₀ → dom 𝐹 = 𝑅)
22	21	3ad2ant3 1132	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) → dom 𝐹 = 𝑅)
23	20, 22	sylan9eqr 2788	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) → dom 𝑟 = 𝑅)
24	23	ad2antrr 723	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) → dom 𝑟 = 𝑅)
25	3	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑟 = 𝐹)
26	25	fveq1d 6886	. . . . . . . . . . . . 13 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑟‘𝑐) = (𝐹‘𝑐))
27	26	breq1d 5151	. . . . . . . . . . . 12 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → ((𝑟‘𝑐) ≤ (♯‘𝑥) ↔ (𝐹‘𝑐) ≤ (♯‘𝑥)))
28	8	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑚 = 𝑀)
29	28	oveq2d 7420	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑚) = (𝑥𝐶𝑀))
30		vex 3472	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
31	12	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑀 ∈ ℕ₀)
32	28, 31	eqeltrd 2827	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑚 ∈ ℕ₀)
33	13	hashbcval 16942	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ V ∧ 𝑚 ∈ ℕ₀) → (𝑥𝐶𝑚) = {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚})
34	30, 32, 33	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑚) = {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚})
35	29, 34	eqtr3d 2768	. . . . . . . . . . . . . 14 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑀) = {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚})
36	35	sseq1d 4008	. . . . . . . . . . . . 13 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → ((𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐}) ↔ {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚} ⊆ (^◡𝑓 “ {𝑐})))
37		rabss 4064	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚} ⊆ (^◡𝑓 “ {𝑐}) ↔ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → 𝑦 ∈ (^◡𝑓 “ {𝑐})))
38	35	eleq2d 2813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑦 ∈ (𝑥𝐶𝑀) ↔ 𝑦 ∈ {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚}))
39		rabid 3446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚} ↔ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚))
40	38, 39	bitrdi 287	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑦 ∈ (𝑥𝐶𝑀) ↔ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)))
41	40	biimpar 477	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)) → 𝑦 ∈ (𝑥𝐶𝑀))
42		elpwi 4604	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝒫 𝑠 → 𝑥 ⊆ 𝑠)
43	42	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → 𝑥 ⊆ 𝑠)
44	13	hashbcss 16944	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑠 ∈ V ∧ 𝑥 ⊆ 𝑠 ∧ 𝑀 ∈ ℕ₀) → (𝑥𝐶𝑀) ⊆ (𝑠𝐶𝑀))
45	11, 43, 31, 44	mp3an2i 1462	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥𝐶𝑀) ⊆ (𝑠𝐶𝑀))
46	45	sselda 3977	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ 𝑦 ∈ (𝑥𝐶𝑀)) → 𝑦 ∈ (𝑠𝐶𝑀))
47	41, 46	syldan 590	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)) → 𝑦 ∈ (𝑠𝐶𝑀))
48		elmapi 8842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀)) → 𝑓:(𝑠𝐶𝑀)⟶𝑅)
49	48	ad3antlr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)) → 𝑓:(𝑠𝐶𝑀)⟶𝑅)
50		ffn 6710	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:(𝑠𝐶𝑀)⟶𝑅 → 𝑓 Fn (𝑠𝐶𝑀))
51		fniniseg 7054	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 Fn (𝑠𝐶𝑀) → (𝑦 ∈ (^◡𝑓 “ {𝑐}) ↔ (𝑦 ∈ (𝑠𝐶𝑀) ∧ (𝑓‘𝑦) = 𝑐)))
52	49, 50, 51	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)) → (𝑦 ∈ (^◡𝑓 “ {𝑐}) ↔ (𝑦 ∈ (𝑠𝐶𝑀) ∧ (𝑓‘𝑦) = 𝑐)))
53	47, 52	mpbirand 704	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚)) → (𝑦 ∈ (^◡𝑓 “ {𝑐}) ↔ (𝑓‘𝑦) = 𝑐))
54	53	anassrs 467	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ 𝑦 ∈ 𝒫 𝑥) ∧ (♯‘𝑦) = 𝑚) → (𝑦 ∈ (^◡𝑓 “ {𝑐}) ↔ (𝑓‘𝑦) = 𝑐))
55	54	pm5.74da 801	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) ∧ 𝑦 ∈ 𝒫 𝑥) → (((♯‘𝑦) = 𝑚 → 𝑦 ∈ (^◡𝑓 “ {𝑐})) ↔ ((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)))
56	55	ralbidva 3169	. . . . . . . . . . . . . 14 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → 𝑦 ∈ (^◡𝑓 “ {𝑐})) ↔ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)))
57	37, 56	bitrid 283	. . . . . . . . . . . . 13 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → ({𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚} ⊆ (^◡𝑓 “ {𝑐}) ↔ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)))
58	36, 57	bitr2d 280	. . . . . . . . . . . 12 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐) ↔ (𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐})))
59	27, 58	anbi12d 630	. . . . . . . . . . 11 ⊢ ((((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) ∧ 𝑥 ∈ 𝒫 𝑠) → (((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)) ↔ ((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐}))))
60	59	rexbidva 3170	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) → (∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)) ↔ ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐}))))
61	24, 60	rexeqbidv 3337	. . . . . . . . 9 ⊢ (((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))) → (∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)) ↔ ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐}))))
62	61	ralbidva 3169	. . . . . . . 8 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → (∀𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)) ↔ ∀𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐}))))
63	18, 62	bitrd 279	. . . . . . 7 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → (∀𝑓 ∈ (dom 𝑟 ↑_m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)) ↔ ∀𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐}))))
64	63	imbi2d 340	. . . . . 6 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → ((𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟 ↑_m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐))) ↔ (𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐})))))
65	64	albidv 1915	. . . . 5 ⊢ ((((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ₀) → (∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟 ↑_m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐))) ↔ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐})))))
66	65	rabbidva 3433	. . . 4 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) → {𝑛 ∈ ℕ₀ ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟 ↑_m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)))} = {𝑛 ∈ ℕ₀ ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐})))})
67		ramval.t	. . . 4 ⊢ 𝑇 = {𝑛 ∈ ℕ₀ ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑_m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (^◡𝑓 “ {𝑐})))}
68	66, 67	eqtr4di 2784	. . 3 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) → {𝑛 ∈ ℕ₀ ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟 ↑_m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)))} = 𝑇)
69	68	infeq1d 9471	. 2 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) → inf({𝑛 ∈ ℕ₀ ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟 ↑_m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)))}, ℝ^, < ) = inf(𝑇, ℝ^, < ))
70		simp1 1133	. 2 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) → 𝑀 ∈ ℕ₀)
71		simp3 1135	. . 3 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) → 𝐹:𝑅⟶ℕ₀)
72		simp2 1134	. . 3 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) → 𝑅 ∈ 𝑉)
73	71, 72	fexd 7223	. 2 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) → 𝐹 ∈ V)
74		xrltso 13123	. . . 4 ⊢ < Or ℝ^*
75	74	infex 9487	. . 3 ⊢ inf(𝑇, ℝ^*, < ) ∈ V
76	75	a1i 11	. 2 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) → inf(𝑇, ℝ^*, < ) ∈ V)
77	2, 69, 70, 73, 76	ovmpod 7555	1 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ₀) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∀wal 1531 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 {crab 3426 Vcvv 3468 ⊆ wss 3943 𝒫 cpw 4597 {csn 4623 class class class wbr 5141 ^◡ccnv 5668 dom cdm 5669 “ cima 5672 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ∈ cmpo 7406 ↑_m cmap 8819 infcinf 9435 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 ℕ₀cn0 12473 ♯chash 14293 Ramsey cram 16939

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-ram 16941

This theorem is referenced by: ramcl2lem 16949

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