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 5905	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝑟 = dom 𝐹)
5		simpll3 1213	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → 𝐹:𝑅⟶ℕ0)
6	5	fdmd 6728	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → dom 𝐹 = 𝑅)
7	4, 6	eqtrd 2771	. . . . . . . . . 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 2742	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → ((♯‘𝑦) = 𝑚 ↔ (♯‘𝑦) = 𝑀))
10	9	rabbidv 3439	. . . . . . . . . . 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 3477	. . . . . . . . . . . 12 ⊢ 𝑠 ∈ V
12		simpll1 1211	. . . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑟 = 𝐹)) ∧ 𝑛 ∈ ℕ0) → {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚} = (𝑠𝐶𝑀))
17	7, 16	oveq12d 7430	. . . . . . . . 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 3324	. . . . . . . 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 5905	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑟 = 𝐹) → dom 𝑟 = dom 𝐹)
21		fdm 6726	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑅⟶ℕ0 → dom 𝐹 = 𝑅)
22	21	3ad2ant3 1134	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → dom 𝐹 = 𝑅)
23	20, 22	sylan9eqr 2793	. . . . . . . . . . 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 6893	. . . . . . . . . . . . 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 5158	. . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . 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 3477	. . . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . 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 4013	. . . . . . . . . . . . 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 4069	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ 𝒫 𝑥 ∣ (♯‘𝑦) = 𝑚} ⊆ (◡𝑓 “ {𝑐}) ↔ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → 𝑦 ∈ (◡𝑓 “ {𝑐})))
38	35	eleq2d 2818	. . . . . . . . . . . . . . . . . . . . 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 3451	. . . . . . . . . . . . . . . . . . . . 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 4609	. . . . . . . . . . . . . . . . . . . . . 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 1465	. . . . . . . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . . . . . . 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 8849	. . . . . . . . . . . . . . . . . . . 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 6717	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:(𝑠𝐶𝑀)⟶𝑅 → 𝑓 Fn (𝑠𝐶𝑀))
51		fniniseg 7061	. . . . . . . . . . . . . . . . . . 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 3174	. . . . . . . . . . . . . 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 3175	. . . . . . . . . 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 3342	. . . . . . . . 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 3174	. . . . . . . 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 1922	. . . . 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 3438	. . . 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 2789	. . 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 9478	. 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 1135	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → 𝑀 ∈ ℕ0)
71		simp3 1137	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → 𝐹:𝑅⟶ℕ0)
72		simp2 1136	. . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → 𝑅 ∈ 𝑉)
73	71, 72	fexd 7231	. 2 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → 𝐹 ∈ V)
74		xrltso 13127	. . . 4 ⊢ < Or ℝ*
75	74	infex 9494	. . 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 7563	1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069  {crab 3431  Vcvv 3473   ⊆ wss 3948  𝒫 cpw 4602  {csn 4628   class class class wbr 5148  ^◡ccnv 5675  dom cdm 5676   “ cima 5679   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412   ∈ cmpo 7414   ↑_m cmap 8826  infcinf 9442  ℝ^*cxr 11254   < clt 11255   ≤ cle 11256  ℕ₀cn0 12479  ♯chash 14297   Ramsey cram 16939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-pre-lttri 11190  ax-pre-lttrn 11191

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-ram 16941

This theorem is referenced by:  ramcl2lem  16949