Theorem ramval 16447

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 16440	. . 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 778	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝑟 = 𝐹)
4	3	dmeqd 5749	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝑟 = dom 𝐹)
5		simpll3 1215	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝐹:𝑅⟶ℕ0)
6	5	fdmd 6516	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝐹 = 𝑅)
7	4, 6	eqtrd 2774	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝑟 = 𝑅)
8		simplrl 777	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝑚 = 𝑀)
9	8	eqeq2d 2750	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → ((♯‘𝑦) = 𝑚 ↔ (♯‘𝑦) = 𝑀))
10	9	rabbidv 3382	. . . . . . . . . . 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 3403	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
12		simpll1 1213	. . . . . . . . . . . 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 16441	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ V ∧ 𝑀 ∈ ℕ0) → (𝑠𝐶𝑀) = {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑀})
15	11, 12, 14	sylancr 590	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → (𝑠𝐶𝑀) = {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑀})
16	10, 15	eqtr4d 2777	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚} = (𝑠𝐶𝑀))
17	7, 16	oveq12d 7191	. . . . . . . . 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 3317	. . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑟 = 𝐹) → 𝑟 = 𝐹)
20	19	dmeqd 5749	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑟 = 𝐹) → dom 𝑟 = dom 𝐹)
21		fdm 6514	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑅⟶ℕ0 → dom 𝐹 = 𝑅)
22	21	3ad2ant3 1136	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → dom 𝐹 = 𝑅)
23	20, 22	sylan9eqr 2796	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) → dom 𝑟 = 𝑅)
24	23	ad2antrr 726	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))) → dom 𝑟 = 𝑅)
25	3	ad2antrr 726	. . . . . . . . . . . . . 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 6679	. . . . . . . . . . . . 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 5041	. . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 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 7189	. . . . . . . . . . . . . . 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 3403	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
31	12	ad2antrr 726	. . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . 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 16441	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ V ∧ 𝑚 ∈ ℕ0) → (𝑥𝐶𝑚) = {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚})
34	30, 32, 33	sylancr 590	. . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . 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 3909	. . . . . . . . . . . . 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 3962	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚} ⊆ (◡𝑓 “ {𝑐}) ↔ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → 𝑦 ∈ (◡𝑓 “ {𝑐})))
38	35	eleq2d 2819	. . . . . . . . . . . . . . . . . . . . 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 3282	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚} ↔ (𝑦 ∈ 𝒫 𝑥 ∧ (♯‘𝑦) = 𝑚))
40	38, 39	bitrdi 290	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 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 4498	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝒫 𝑠 → 𝑥 ⊆ 𝑠)
43	42	adantl 485	. . . . . . . . . . . . . . . . . . . . 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 16443	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑠 ∈ V ∧ 𝑥 ⊆ 𝑠 ∧ 𝑀 ∈ ℕ0) → (𝑥𝐶𝑀) ⊆ (𝑠𝐶𝑀))
45	11, 43, 31, 44	mp3an2i 1467	. . . . . . . . . . . . . . . . . . . 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 3878	. . . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . 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 8462	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀)) → 𝑓:(𝑠𝐶𝑀)⟶𝑅)
49	48	ad3antlr 731	. . . . . . . . . . . . . . . . . . 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 6505	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:(𝑠𝐶𝑀)⟶𝑅 → 𝑓 Fn (𝑠𝐶𝑀))
51		fniniseg 6840	. . . . . . . . . . . . . . . . . . 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 707	. . . . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . . 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 804	. . . . . . . . . . . . . . 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 3109	. . . . . . . . . . . . . 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	syl5bb 286	. . . . . . . . . . . . 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 283	. . . . . . . . . . . 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 634	. . . . . . . . . . 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 3207	. . . . . . . . . 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 3306	. . . . . . . . 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 3109	. . . . . . . 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 282	. . . . . . 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 344	. . . . . 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 1927	. . . . 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 3380	. . . 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 2792	. . 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 9017	. 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 1137	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → 𝑀 ∈ ℕ0)
71		simp3 1139	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → 𝐹:𝑅⟶ℕ0)
72		simp2 1138	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → 𝑅 ∈ 𝑉)
73	71, 72	fexd 7003	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → 𝐹 ∈ V)
74		xrltso 12620	. . . 4 ⊢ < Or ℝ*
75	74	infex 9033	. . 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 7320	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3054  ∃wrex 3055  {crab 3058  Vcvv 3399   ⊆ wss 3844  𝒫 cpw 4489  {csn 4517   class class class wbr 5031  ^◡ccnv 5525  dom cdm 5526   “ cima 5529   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7173   ∈ cmpo 7175   ↑_m cmap 8440  infcinf 8981  ℝ^*cxr 10755   < clt 10756   ≤ cle 10757  ℕ₀cn0 11979  ♯chash 13785   Ramsey cram 16438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482  ax-cnex 10674  ax-resscn 10675  ax-pre-lttri 10692  ax-pre-lttrn 10693

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-po 5443  df-so 5444  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-1st 7717  df-2nd 7718  df-er 8323  df-map 8442  df-en 8559  df-dom 8560  df-sdom 8561  df-sup 8982  df-inf 8983  df-pnf 10758  df-mnf 10759  df-xr 10760  df-ltxr 10761  df-ram 16440

This theorem is referenced by:  ramcl2lem  16448