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Theorem ramub1lem2 16418

Description: Lemma for ramub1 16419. (Contributed by Mario Carneiro, 23-Apr-2015.)

**Hypotheses**
Ref	Expression
ramub1.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
ramub1.r	⊢ (𝜑 → 𝑅 ∈ Fin)
ramub1.f	⊢ (𝜑 → 𝐹:𝑅⟶ℕ)
ramub1.g	⊢ 𝐺 = (𝑥 ∈ 𝑅 ↦ (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦)))))
ramub1.1	⊢ (𝜑 → 𝐺:𝑅⟶ℕ₀)
ramub1.2	⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ₀)
ramub1.3	⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
ramub1.4	⊢ (𝜑 → 𝑆 ∈ Fin)
ramub1.5	⊢ (𝜑 → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))
ramub1.6	⊢ (𝜑 → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
ramub1.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)
ramub1.h	⊢ 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))

**Assertion**
Ref	Expression
ramub1lem2	⊢ (𝜑 → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐})))

Distinct variable groups: 𝑥,𝑢,𝑐,𝑦,𝑧,𝐹 𝑎,𝑏,𝑐,𝑖,𝑢,𝑥,𝑦,𝑧,𝑀 𝐺,𝑎,𝑐,𝑖,𝑢,𝑥,𝑦,𝑧 𝑅,𝑐,𝑢,𝑥,𝑦,𝑧 𝜑,𝑐,𝑢,𝑥,𝑦,𝑧 𝑆,𝑎,𝑐,𝑖,𝑢,𝑥,𝑦,𝑧 𝐶,𝑐,𝑢,𝑥,𝑦,𝑧 𝐻,𝑐,𝑢,𝑥,𝑦,𝑧 𝐾,𝑐,𝑢,𝑥,𝑦,𝑧 𝑋,𝑎,𝑐,𝑖,𝑢,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑖,𝑎,𝑏) 𝐶(𝑖,𝑎,𝑏) 𝑅(𝑖,𝑎,𝑏) 𝑆(𝑏) 𝐹(𝑖,𝑎,𝑏) 𝐺(𝑏) 𝐻(𝑖,𝑎,𝑏) 𝐾(𝑖,𝑎,𝑏) 𝑋(𝑏)

Proof of Theorem ramub1lem2 Dummy variables 𝑑 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ramub1.3	. . 3 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ₀ ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
2		ramub1.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		nnm1nn0 11975	. . . 4 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ₀)
5		ramub1.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Fin)
6		ramub1.1	. . 3 ⊢ (𝜑 → 𝐺:𝑅⟶ℕ₀)
7		ramub1.2	. . 3 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ₀)
8		ramub1.4	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Fin)
9		diffi 8786	. . . 4 ⊢ (𝑆 ∈ Fin → (𝑆 ∖ {𝑋}) ∈ Fin)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → (𝑆 ∖ {𝑋}) ∈ Fin)
11	7	nn0red 11995	. . . . 5 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℝ)
12	11	leidd 11244	. . . 4 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ≤ ((𝑀 − 1) Ramsey 𝐺))
13		hashcl 13767	. . . . . . 7 ⊢ ((𝑆 ∖ {𝑋}) ∈ Fin → (♯‘(𝑆 ∖ {𝑋})) ∈ ℕ₀)
14	10, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) ∈ ℕ₀)
15	14	nn0cnd 11996	. . . . 5 ⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) ∈ ℂ)
16	7	nn0cnd 11996	. . . . 5 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℂ)
17		1cnd 10674	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
18		undif1 4372	. . . . . . . 8 ⊢ ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = (𝑆 ∪ {𝑋})
19		ramub1.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
20	19	snssd 4699	. . . . . . . . 9 ⊢ (𝜑 → {𝑋} ⊆ 𝑆)
21		ssequn2 4088	. . . . . . . . 9 ⊢ ({𝑋} ⊆ 𝑆 ↔ (𝑆 ∪ {𝑋}) = 𝑆)
22	20, 21	sylib 221	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {𝑋}) = 𝑆)
23	18, 22	syl5eq 2805	. . . . . . 7 ⊢ (𝜑 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
24	23	fveq2d 6662	. . . . . 6 ⊢ (𝜑 → (♯‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = (♯‘𝑆))
25		neldifsnd 4683	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
26		hashunsng 13803	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑆 → (((𝑆 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) → (♯‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑆 ∖ {𝑋})) + 1)))
27	19, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → (((𝑆 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) → (♯‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑆 ∖ {𝑋})) + 1)))
28	10, 25, 27	mp2and 698	. . . . . 6 ⊢ (𝜑 → (♯‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑆 ∖ {𝑋})) + 1))
29		ramub1.5	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))
30	24, 28, 29	3eqtr3d 2801	. . . . 5 ⊢ (𝜑 → ((♯‘(𝑆 ∖ {𝑋})) + 1) = (((𝑀 − 1) Ramsey 𝐺) + 1))
31	15, 16, 17, 30	addcan2ad 10884	. . . 4 ⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) = ((𝑀 − 1) Ramsey 𝐺))
32	12, 31	breqtrrd 5060	. . 3 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ≤ (♯‘(𝑆 ∖ {𝑋})))
33		ramub1.6	. . . . . 6 ⊢ (𝜑 → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
34	33	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
35		fveqeq2 6667	. . . . . . 7 ⊢ (𝑥 = (𝑢 ∪ {𝑋}) → ((♯‘𝑥) = 𝑀 ↔ (♯‘(𝑢 ∪ {𝑋})) = 𝑀))
36	1	hashbcval 16393	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∖ {𝑋}) ∈ Fin ∧ (𝑀 − 1) ∈ ℕ₀) → ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) = {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)})
37	10, 4, 36	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) = {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)})
38	37	eleq2d 2837	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↔ 𝑢 ∈ {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)}))
39		fveqeq2 6667	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → ((♯‘𝑥) = (𝑀 − 1) ↔ (♯‘𝑢) = (𝑀 − 1)))
40	39	elrab 3602	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)} ↔ (𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∧ (♯‘𝑢) = (𝑀 − 1)))
41	38, 40	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↔ (𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∧ (♯‘𝑢) = (𝑀 − 1))))
42	41	simprbda 502	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}))
43	42	elpwid 4505	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ⊆ (𝑆 ∖ {𝑋}))
44	43	difss2d 4040	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ⊆ 𝑆)
45	20	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → {𝑋} ⊆ 𝑆)
46	44, 45	unssd 4091	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ⊆ 𝑆)
47		vex 3413	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
48		snex 5300	. . . . . . . . . 10 ⊢ {𝑋} ∈ V
49	47, 48	unex 7467	. . . . . . . . 9 ⊢ (𝑢 ∪ {𝑋}) ∈ V
50	49	elpw 4498	. . . . . . . 8 ⊢ ((𝑢 ∪ {𝑋}) ∈ 𝒫 𝑆 ↔ (𝑢 ∪ {𝑋}) ⊆ 𝑆)
51	46, 50	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ 𝒫 𝑆)
52	10	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑆 ∖ {𝑋}) ∈ Fin)
53	52, 43	ssfid 8778	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ∈ Fin)
54		neldifsnd 4683	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
55	43, 54	ssneldd 3895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ¬ 𝑋 ∈ 𝑢)
56	19	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑋 ∈ 𝑆)
57		hashunsng 13803	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑆 → ((𝑢 ∈ Fin ∧ ¬ 𝑋 ∈ 𝑢) → (♯‘(𝑢 ∪ {𝑋})) = ((♯‘𝑢) + 1)))
58	56, 57	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((𝑢 ∈ Fin ∧ ¬ 𝑋 ∈ 𝑢) → (♯‘(𝑢 ∪ {𝑋})) = ((♯‘𝑢) + 1)))
59	53, 55, 58	mp2and 698	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (♯‘(𝑢 ∪ {𝑋})) = ((♯‘𝑢) + 1))
60	41	simplbda 503	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (♯‘𝑢) = (𝑀 − 1))
61	60	oveq1d 7165	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((♯‘𝑢) + 1) = ((𝑀 − 1) + 1))
62	2	nncnd 11690	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
63		ax-1cn 10633	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
64		npcan 10933	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀)
65	62, 63, 64	sylancl 589	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
66	65	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
67	59, 61, 66	3eqtrd 2797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (♯‘(𝑢 ∪ {𝑋})) = 𝑀)
68	35, 51, 67	elrabd 3604	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
69	2	nnnn0d 11994	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
70	1	hashbcval 16393	. . . . . . . 8 ⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ ℕ₀) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
71	8, 69, 70	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
72	71	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
73	68, 72	eleqtrrd 2855	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ (𝑆𝐶𝑀))
74	34, 73	ffvelrnd 6843	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝐾‘(𝑢 ∪ {𝑋})) ∈ 𝑅)
75		ramub1.h	. . . 4 ⊢ 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))
76	74, 75	fmptd 6869	. . 3 ⊢ (𝜑 → 𝐻:((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))⟶𝑅)
77	1, 4, 5, 6, 7, 10, 32, 76	rami 16406	. 2 ⊢ (𝜑 → ∃𝑑 ∈ 𝑅 ∃𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))
78	69	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑀 ∈ ℕ₀)
79	5	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑅 ∈ Fin)
80		ramub1.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑅⟶ℕ)
81	80	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝐹:𝑅⟶ℕ)
82		simprll 778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑑 ∈ 𝑅)
83	81, 82	ffvelrnd 6843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝐹‘𝑑) ∈ ℕ)
84		nnm1nn0 11975	. . . . . . . . . 10 ⊢ ((𝐹‘𝑑) ∈ ℕ → ((𝐹‘𝑑) − 1) ∈ ℕ₀)
85	83, 84	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → ((𝐹‘𝑑) − 1) ∈ ℕ₀)
86	85	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ 𝑦 ∈ 𝑅) → ((𝐹‘𝑑) − 1) ∈ ℕ₀)
87	81	ffvelrnda 6842	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ 𝑦 ∈ 𝑅) → (𝐹‘𝑦) ∈ ℕ)
88	87	nnnn0d 11994	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ 𝑦 ∈ 𝑅) → (𝐹‘𝑦) ∈ ℕ₀)
89	86, 88	ifcld 4466	. . . . . . 7 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ 𝑦 ∈ 𝑅) → if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)) ∈ ℕ₀)
90		eqid 2758	. . . . . . 7 ⊢ (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦))) = (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))
91	89, 90	fmptd 6869	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦))):𝑅⟶ℕ₀)
92		equequ2 2033	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑑 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑑))
93		fveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑑 → (𝐹‘𝑥) = (𝐹‘𝑑))
94	93	oveq1d 7165	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑑 → ((𝐹‘𝑥) − 1) = ((𝐹‘𝑑) − 1))
95	92, 94	ifbieq1d 4444	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑑 → if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦)) = if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))
96	95	mpteq2dv 5128	. . . . . . . . . 10 ⊢ (𝑥 = 𝑑 → (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦))) = (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦))))
97	96	oveq2d 7166	. . . . . . . . 9 ⊢ (𝑥 = 𝑑 → (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦)))) = (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))))
98		ramub1.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑥 ∈ 𝑅 ↦ (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦)))))
99		ovex 7183	. . . . . . . . 9 ⊢ (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))) ∈ V
100	97, 98, 99	fvmpt 6759	. . . . . . . 8 ⊢ (𝑑 ∈ 𝑅 → (𝐺‘𝑑) = (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))))
101	82, 100	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝐺‘𝑑) = (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))))
102	6	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝐺:𝑅⟶ℕ₀)
103	102, 82	ffvelrnd 6843	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝐺‘𝑑) ∈ ℕ₀)
104	101, 103	eqeltrrd 2853	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))) ∈ ℕ₀)
105		simprlr 779	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋}))
106		simprrl 780	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝐺‘𝑑) ≤ (♯‘𝑤))
107	101, 106	eqbrtrrd 5056	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))) ≤ (♯‘𝑤))
108	33	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
109	8	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑆 ∈ Fin)
110	105	elpwid 4505	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑤 ⊆ (𝑆 ∖ {𝑋}))
111	110	difss2d 4040	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑤 ⊆ 𝑆)
112	1	hashbcss 16395	. . . . . . . 8 ⊢ ((𝑆 ∈ Fin ∧ 𝑤 ⊆ 𝑆 ∧ 𝑀 ∈ ℕ₀) → (𝑤𝐶𝑀) ⊆ (𝑆𝐶𝑀))
113	109, 111, 78, 112	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝑤𝐶𝑀) ⊆ (𝑆𝐶𝑀))
114	108, 113	fssresd 6530	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝐾 ↾ (𝑤𝐶𝑀)):(𝑤𝐶𝑀)⟶𝑅)
115	1, 78, 79, 91, 104, 105, 107, 114	rami 16406	. . . . 5 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → ∃𝑐 ∈ 𝑅 ∃𝑣 ∈ 𝒫 𝑤(((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))
116		equequ1 2032	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑐 → (𝑦 = 𝑑 ↔ 𝑐 = 𝑑))
117		fveq2 6658	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑐 → (𝐹‘𝑦) = (𝐹‘𝑐))
118	116, 117	ifbieq2d 4446	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑐 → if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)) = if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)))
119		ovex 7183	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑑) − 1) ∈ V
120		fvex 6671	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑐) ∈ V
121	119, 120	ifex 4470	. . . . . . . . . . . . 13 ⊢ if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ∈ V
122	118, 90, 121	fvmpt 6759	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝑅 → ((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) = if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)))
123	122	ad2antrl 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ (𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤)) → ((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) = if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)))
124	123	breq1d 5042	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ (𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤)) → (((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ↔ if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣)))
125	124	anbi1d 632	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ (𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤)) → ((((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) ↔ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}))))
126	2	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑀 ∈ ℕ)
127	5	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑅 ∈ Fin)
128	80	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝐹:𝑅⟶ℕ)
129	6	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝐺:𝑅⟶ℕ₀)
130	7	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ₀)
131	8	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑆 ∈ Fin)
132	29	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))
133	33	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
134	19	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑋 ∈ 𝑆)
135	82	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑑 ∈ 𝑅)
136	110	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑤 ⊆ (𝑆 ∖ {𝑋}))
137	106	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝐺‘𝑑) ≤ (♯‘𝑤))
138		simprrr 781	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑}))
139	138	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑}))
140		simprll 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑐 ∈ 𝑅)
141		simprlr 779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑣 ∈ 𝒫 𝑤)
142	141	elpwid 4505	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑣 ⊆ 𝑤)
143		simprrl 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣))
144		simprrr 781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}))
145		cnvresima 6059	. . . . . . . . . . . . 13 ⊢ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}) = ((^◡𝐾 “ {𝑐}) ∩ (𝑤𝐶𝑀))
146		inss1 4133	. . . . . . . . . . . . 13 ⊢ ((^◡𝐾 “ {𝑐}) ∩ (𝑤𝐶𝑀)) ⊆ (^◡𝐾 “ {𝑐})
147	145, 146	eqsstri 3926	. . . . . . . . . . . 12 ⊢ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}) ⊆ (^◡𝐾 “ {𝑐})
148	144, 147	sstrdi 3904	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝑣𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐}))
149	126, 127, 128, 98, 129, 130, 1, 131, 132, 133, 134, 75, 135, 136, 137, 139, 140, 142, 143, 148	ramub1lem1 16417	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ ((𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐})))
150	149	expr 460	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ (𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤)) → ((if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐}))))
151	125, 150	sylbid 243	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ (𝑐 ∈ 𝑅 ∧ 𝑣 ∈ 𝒫 𝑤)) → ((((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐}))))
152	151	anassrs 471	. . . . . . 7 ⊢ ((((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ 𝑐 ∈ 𝑅) ∧ 𝑣 ∈ 𝒫 𝑤) → ((((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐}))))
153	152	rexlimdva 3208	. . . . . 6 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ 𝑐 ∈ 𝑅) → (∃𝑣 ∈ 𝒫 𝑤(((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐}))))
154	153	reximdva 3198	. . . . 5 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (∃𝑐 ∈ 𝑅 ∃𝑣 ∈ 𝒫 𝑤(((𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))‘𝑐) ≤ (♯‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐}))))
155	115, 154	mpd 15	. . . 4 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐})))
156	155	expr 460	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋}))) → (((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})) → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐}))))
157	156	rexlimdvva 3218	. 2 ⊢ (𝜑 → (∃𝑑 ∈ 𝑅 ∃𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})) → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐}))))
158	77, 157	mpd 15	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (^◡𝐾 “ {𝑐})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3071 {crab 3074 Vcvv 3409 ∖ cdif 3855 ∪ cun 3856 ∩ cin 3857 ⊆ wss 3858 ifcif 4420 𝒫 cpw 4494 {csn 4522 class class class wbr 5032 ↦ cmpt 5112 ^◡ccnv 5523 ↾ cres 5526 “ cima 5527 ⟶wf 6331 ‘cfv 6335 (class class class)co 7150 ∈ cmpo 7152 Fincfn 8527 ℂcc 10573 1c1 10576 + caddc 10578 ≤ cle 10714 − cmin 10908 ℕcn 11674 ℕ₀cn0 11934 ♯chash 13740 Ramsey cram 16390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7580 df-1st 7693 df-2nd 7694 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-oadd 8116 df-er 8299 df-map 8418 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-sup 8939 df-inf 8940 df-dju 9363 df-card 9401 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-nn 11675 df-n0 11935 df-z 12021 df-uz 12283 df-fz 12940 df-hash 13741 df-ram 16392

This theorem is referenced by: ramub1 16419

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