Theorem ramub2 16350

Description: It is sufficient to check the Ramsey property on finite sets of size equal to the upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
rami.c	⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
rami.m	⊢ (𝜑 → 𝑀 ∈ ℕ0)
rami.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
rami.f	⊢ (𝜑 → 𝐹:𝑅⟶ℕ0)
ramub2.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
ramub2.i	⊢ ((𝜑 ∧ ((♯‘𝑠) = 𝑁 ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))

Assertion

Ref	Expression
ramub2	⊢ (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁)

Distinct variable groups:   𝑓,𝑐,𝑠,𝑥,𝐶   𝜑,𝑐,𝑓,𝑠,𝑥   𝐹,𝑐,𝑓,𝑠,𝑥   𝑎,𝑏,𝑐,𝑓,𝑖,𝑠,𝑥,𝑀   𝑅,𝑐,𝑓,𝑠,𝑥   𝑁,𝑎,𝑐,𝑓,𝑖,𝑠,𝑥   𝑉,𝑐,𝑓,𝑠,𝑥

Allowed substitution hints:   𝜑(𝑖,𝑎,𝑏)   𝐶(𝑖,𝑎,𝑏)   𝑅(𝑖,𝑎,𝑏)   𝐹(𝑖,𝑎,𝑏)   𝑁(𝑏)   𝑉(𝑖,𝑎,𝑏)

Proof of Theorem ramub2

Dummy variables 𝑔 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rami.c	. 2 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
2		rami.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
3		rami.r	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
4		rami.f	. 2 ⊢ (𝜑 → 𝐹:𝑅⟶ℕ0)
5		ramub2.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
6	5	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) → 𝑁 ∈ ℕ0)
7		hashfz1 13707	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
8	6, 7	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) → (♯‘(1...𝑁)) = 𝑁)
9		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) → 𝑁 ≤ (♯‘𝑡))
10	8, 9	eqbrtrd 5088	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) → (♯‘(1...𝑁)) ≤ (♯‘𝑡))
11		fzfid 13342	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) → (1...𝑁) ∈ Fin)
12		vex 3497	. . . . . 6 ⊢ 𝑡 ∈ V
13		hashdom 13741	. . . . . 6 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑡 ∈ V) → ((♯‘(1...𝑁)) ≤ (♯‘𝑡) ↔ (1...𝑁) ≼ 𝑡))
14	11, 12, 13	sylancl 588	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) → ((♯‘(1...𝑁)) ≤ (♯‘𝑡) ↔ (1...𝑁) ≼ 𝑡))
15	10, 14	mpbid 234	. . . 4 ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) → (1...𝑁) ≼ 𝑡)
16	12	domen 8522	. . . 4 ⊢ ((1...𝑁) ≼ 𝑡 ↔ ∃𝑠((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡))
17	15, 16	sylib 220	. . 3 ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) → ∃𝑠((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡))
18		simpll 765	. . . . 5 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → 𝜑)
19		ensym 8558	. . . . . . . 8 ⊢ ((1...𝑁) ≈ 𝑠 → 𝑠 ≈ (1...𝑁))
20	19	ad2antrl 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → 𝑠 ≈ (1...𝑁))
21		hasheni 13709	. . . . . . 7 ⊢ (𝑠 ≈ (1...𝑁) → (♯‘𝑠) = (♯‘(1...𝑁)))
22	20, 21	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → (♯‘𝑠) = (♯‘(1...𝑁)))
23	5	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → 𝑁 ∈ ℕ0)
24	23, 7	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → (♯‘(1...𝑁)) = 𝑁)
25	22, 24	eqtrd 2856	. . . . 5 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → (♯‘𝑠) = 𝑁)
26		simplrr 776	. . . . . 6 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → 𝑔:(𝑡𝐶𝑀)⟶𝑅)
27		simprr 771	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → 𝑠 ⊆ 𝑡)
28	2	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → 𝑀 ∈ ℕ0)
29	1	hashbcss 16340	. . . . . . 7 ⊢ ((𝑡 ∈ V ∧ 𝑠 ⊆ 𝑡 ∧ 𝑀 ∈ ℕ0) → (𝑠𝐶𝑀) ⊆ (𝑡𝐶𝑀))
30	12, 27, 28, 29	mp3an2i 1462	. . . . . 6 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → (𝑠𝐶𝑀) ⊆ (𝑡𝐶𝑀))
31	26, 30	fssresd 6545	. . . . 5 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → (𝑔 ↾ (𝑠𝐶𝑀)):(𝑠𝐶𝑀)⟶𝑅)
32		vex 3497	. . . . . . 7 ⊢ 𝑔 ∈ V
33	32	resex 5899	. . . . . 6 ⊢ (𝑔 ↾ (𝑠𝐶𝑀)) ∈ V
34		feq1 6495	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ↾ (𝑠𝐶𝑀)) → (𝑓:(𝑠𝐶𝑀)⟶𝑅 ↔ (𝑔 ↾ (𝑠𝐶𝑀)):(𝑠𝐶𝑀)⟶𝑅))
35	34	anbi2d 630	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ↾ (𝑠𝐶𝑀)) → (((♯‘𝑠) = 𝑁 ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅) ↔ ((♯‘𝑠) = 𝑁 ∧ (𝑔 ↾ (𝑠𝐶𝑀)):(𝑠𝐶𝑀)⟶𝑅)))
36	35	anbi2d 630	. . . . . . 7 ⊢ (𝑓 = (𝑔 ↾ (𝑠𝐶𝑀)) → ((𝜑 ∧ ((♯‘𝑠) = 𝑁 ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) ↔ (𝜑 ∧ ((♯‘𝑠) = 𝑁 ∧ (𝑔 ↾ (𝑠𝐶𝑀)):(𝑠𝐶𝑀)⟶𝑅))))
37		cnveq 5744	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔 ↾ (𝑠𝐶𝑀)) → ◡𝑓 = ◡(𝑔 ↾ (𝑠𝐶𝑀)))
38	37	imaeq1d 5928	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ↾ (𝑠𝐶𝑀)) → (◡𝑓 “ {𝑐}) = (◡(𝑔 ↾ (𝑠𝐶𝑀)) “ {𝑐}))
39		cnvresima 6087	. . . . . . . . . . 11 ⊢ (◡(𝑔 ↾ (𝑠𝐶𝑀)) “ {𝑐}) = ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀))
40	38, 39	syl6eq 2872	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ↾ (𝑠𝐶𝑀)) → (◡𝑓 “ {𝑐}) = ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀)))
41	40	sseq2d 3999	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ↾ (𝑠𝐶𝑀)) → ((𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}) ↔ (𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀))))
42	41	anbi2d 630	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ↾ (𝑠𝐶𝑀)) → (((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})) ↔ ((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀)))))
43	42	2rexbidv 3300	. . . . . . 7 ⊢ (𝑓 = (𝑔 ↾ (𝑠𝐶𝑀)) → (∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})) ↔ ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀)))))
44	36, 43	imbi12d 347	. . . . . 6 ⊢ (𝑓 = (𝑔 ↾ (𝑠𝐶𝑀)) → (((𝜑 ∧ ((♯‘𝑠) = 𝑁 ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) ↔ ((𝜑 ∧ ((♯‘𝑠) = 𝑁 ∧ (𝑔 ↾ (𝑠𝐶𝑀)):(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀))))))
45		ramub2.i	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝑠) = 𝑁 ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))
46	33, 44, 45	vtocl 3559	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝑠) = 𝑁 ∧ (𝑔 ↾ (𝑠𝐶𝑀)):(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀))))
47	18, 25, 31, 46	syl12anc 834	. . . 4 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀))))
48		sstr 3975	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑡) → 𝑥 ⊆ 𝑡)
49	48	expcom 416	. . . . . . . . 9 ⊢ (𝑠 ⊆ 𝑡 → (𝑥 ⊆ 𝑠 → 𝑥 ⊆ 𝑡))
50	49	ad2antll 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → (𝑥 ⊆ 𝑠 → 𝑥 ⊆ 𝑡))
51		velpw 4544	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝑠 ↔ 𝑥 ⊆ 𝑠)
52		velpw 4544	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝑡 ↔ 𝑥 ⊆ 𝑡)
53	50, 51, 52	3imtr4g 298	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → (𝑥 ∈ 𝒫 𝑠 → 𝑥 ∈ 𝒫 𝑡))
54		id 22	. . . . . . . . . 10 ⊢ ((𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀)) → (𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀)))
55		inss1 4205	. . . . . . . . . 10 ⊢ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀)) ⊆ (◡𝑔 “ {𝑐})
56	54, 55	sstrdi 3979	. . . . . . . . 9 ⊢ ((𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀)) → (𝑥𝐶𝑀) ⊆ (◡𝑔 “ {𝑐}))
57	56	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → ((𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀)) → (𝑥𝐶𝑀) ⊆ (◡𝑔 “ {𝑐})))
58	57	anim2d 613	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → (((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀))) → ((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑔 “ {𝑐}))))
59	53, 58	anim12d 610	. . . . . 6 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → ((𝑥 ∈ 𝒫 𝑠 ∧ ((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀)))) → (𝑥 ∈ 𝒫 𝑡 ∧ ((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑔 “ {𝑐})))))
60	59	reximdv2 3271	. . . . 5 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → (∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀))) → ∃𝑥 ∈ 𝒫 𝑡((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑔 “ {𝑐}))))
61	60	reximdv 3273	. . . 4 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → (∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ ((◡𝑔 “ {𝑐}) ∩ (𝑠𝐶𝑀))) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑡((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑔 “ {𝑐}))))
62	47, 61	mpd 15	. . 3 ⊢ (((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) ∧ ((1...𝑁) ≈ 𝑠 ∧ 𝑠 ⊆ 𝑡)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑡((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑔 “ {𝑐})))
63	17, 62	exlimddv 1936	. 2 ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑡) ∧ 𝑔:(𝑡𝐶𝑀)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑡((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑔 “ {𝑐})))
64	1, 2, 3, 4, 5, 63	ramub 16349	1 ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃wrex 3139  {crab 3142  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936  𝒫 cpw 4539  {csn 4567   class class class wbr 5066  ^◡ccnv 5554   ↾ cres 5557   “ cima 5558  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158   ≈ cen 8506   ≼ cdom 8507  Fincfn 8509  1c1 10538   ≤ cle 10676  ℕ₀cn0 11898  ...cfz 12893  ♯chash 13691   Ramsey cram 16335

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-inf 8907  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-xnn0 11969  df-z 11983  df-uz 12245  df-fz 12894  df-hash 13692  df-ram 16337

This theorem is referenced by:  ramub1  16364