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Theorem ramub1lem2 16353

Description: Lemma for ramub1 16354. (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 11926	. . . 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 8734	. . . 4 ⊢ (𝑆 ∈ Fin → (𝑆 ∖ {𝑋}) ∈ Fin)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → (𝑆 ∖ {𝑋}) ∈ Fin)
11	7	nn0red 11944	. . . . 5 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℝ)
12	11	leidd 11195	. . . 4 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ≤ ((𝑀 − 1) Ramsey 𝐺))
13		hashcl 13713	. . . . . . 7 ⊢ ((𝑆 ∖ {𝑋}) ∈ Fin → (♯‘(𝑆 ∖ {𝑋})) ∈ ℕ₀)
14	10, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) ∈ ℕ₀)
15	14	nn0cnd 11945	. . . . 5 ⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) ∈ ℂ)
16	7	nn0cnd 11945	. . . . 5 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℂ)
17		1cnd 10625	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
18		undif1 4382	. . . . . . . 8 ⊢ ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = (𝑆 ∪ {𝑋})
19		ramub1.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
20	19	snssd 4702	. . . . . . . . 9 ⊢ (𝜑 → {𝑋} ⊆ 𝑆)
21		ssequn2 4110	. . . . . . . . 9 ⊢ ({𝑋} ⊆ 𝑆 ↔ (𝑆 ∪ {𝑋}) = 𝑆)
22	20, 21	sylib 221	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {𝑋}) = 𝑆)
23	18, 22	syl5eq 2845	. . . . . . 7 ⊢ (𝜑 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
24	23	fveq2d 6649	. . . . . 6 ⊢ (𝜑 → (♯‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = (♯‘𝑆))
25		neldifsnd 4686	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
26		hashunsng 13749	. . . . . . . 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 2841	. . . . 5 ⊢ (𝜑 → ((♯‘(𝑆 ∖ {𝑋})) + 1) = (((𝑀 − 1) Ramsey 𝐺) + 1))
31	15, 16, 17, 30	addcan2ad 10835	. . . 4 ⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) = ((𝑀 − 1) Ramsey 𝐺))
32	12, 31	breqtrrd 5058	. . 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 6654	. . . . . . 7 ⊢ (𝑥 = (𝑢 ∪ {𝑋}) → ((♯‘𝑥) = 𝑀 ↔ (♯‘(𝑢 ∪ {𝑋})) = 𝑀))
36	1	hashbcval 16328	. . . . . . . . . . . . . . 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 2875	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↔ 𝑢 ∈ {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)}))
39		fveqeq2 6654	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → ((♯‘𝑥) = (𝑀 − 1) ↔ (♯‘𝑢) = (𝑀 − 1)))
40	39	elrab 3628	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)} ↔ (𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∧ (♯‘𝑢) = (𝑀 − 1)))
41	38, 40	syl6bb 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 4508	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ⊆ (𝑆 ∖ {𝑋}))
44	43	difss2d 4062	. . . . . . . . 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 4113	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ⊆ 𝑆)
47		vex 3444	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
48		snex 5297	. . . . . . . . . 10 ⊢ {𝑋} ∈ V
49	47, 48	unex 7449	. . . . . . . . 9 ⊢ (𝑢 ∪ {𝑋}) ∈ V
50	49	elpw 4501	. . . . . . . 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 8725	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ∈ Fin)
54		neldifsnd 4686	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
55	43, 54	ssneldd 3918	. . . . . . . . 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 13749	. . . . . . . . . 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 7150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((♯‘𝑢) + 1) = ((𝑀 − 1) + 1))
62	2	nncnd 11641	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
63		ax-1cn 10584	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
64		npcan 10884	. . . . . . . . . 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 2837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (♯‘(𝑢 ∪ {𝑋})) = 𝑀)
68	35, 51, 67	elrabd 3630	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
69	2	nnnn0d 11943	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
70	1	hashbcval 16328	. . . . . . . 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 2893	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ (𝑆𝐶𝑀))
74	34, 73	ffvelrnd 6829	. . . 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 6855	. . 3 ⊢ (𝜑 → 𝐻:((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))⟶𝑅)
77	1, 4, 5, 6, 7, 10, 32, 76	rami 16341	. 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 6829	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝐹‘𝑑) ∈ ℕ)
84		nnm1nn0 11926	. . . . . . . . . 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 6828	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ 𝑦 ∈ 𝑅) → (𝐹‘𝑦) ∈ ℕ)
88	87	nnnn0d 11943	. . . . . . . 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 4470	. . . . . . 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 2798	. . . . . . 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 6855	. . . . . 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 6645	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑑 → (𝐹‘𝑥) = (𝐹‘𝑑))
94	93	oveq1d 7150	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑑 → ((𝐹‘𝑥) − 1) = ((𝐹‘𝑑) − 1))
95	92, 94	ifbieq1d 4448	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑑 → if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦)) = if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))
96	95	mpteq2dv 5126	. . . . . . . . . 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 7151	. . . . . . . . 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 7168	. . . . . . . . 9 ⊢ (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))) ∈ V
100	97, 98, 99	fvmpt 6745	. . . . . . . 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 6829	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝐺‘𝑑) ∈ ℕ₀)
104	101, 103	eqeltrrd 2891	. . . . . 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 5054	. . . . . 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 4508	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑤 ⊆ (𝑆 ∖ {𝑋}))
111	110	difss2d 4062	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑤 ⊆ 𝑆)
112	1	hashbcss 16330	. . . . . . . 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 6519	. . . . . 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 16341	. . . . 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 6645	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑐 → (𝐹‘𝑦) = (𝐹‘𝑐))
118	116, 117	ifbieq2d 4450	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑐 → if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)) = if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)))
119		ovex 7168	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑑) − 1) ∈ V
120		fvex 6658	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑐) ∈ V
121	119, 120	ifex 4473	. . . . . . . . . . . . 13 ⊢ if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ∈ V
122	118, 90, 121	fvmpt 6745	. . . . . . . . . . . 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 5040	. . . . . . . . . 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 4508	. . . . . . . . . . 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 6054	. . . . . . . . . . . . 13 ⊢ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}) = ((^◡𝐾 “ {𝑐}) ∩ (𝑤𝐶𝑀))
146		inss1 4155	. . . . . . . . . . . . 13 ⊢ ((^◡𝐾 “ {𝑐}) ∩ (𝑤𝐶𝑀)) ⊆ (^◡𝐾 “ {𝑐})
147	145, 146	eqsstri 3949	. . . . . . . . . . . 12 ⊢ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}) ⊆ (^◡𝐾 “ {𝑐})
148	144, 147	sstrdi 3927	. . . . . . . . . . 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 16352	. . . . . . . . . 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 3243	. . . . . 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 3233	. . . . 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 3253	. 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 3107 {crab 3110 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ifcif 4425 𝒫 cpw 4497 {csn 4525 class class class wbr 5030 ↦ cmpt 5110 ^◡ccnv 5518 ↾ cres 5521 “ cima 5522 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 Fincfn 8492 ℂcc 10524 1c1 10527 + caddc 10529 ≤ cle 10665 − cmin 10859 ℕcn 11625 ℕ₀cn0 11885 ♯chash 13686 Ramsey cram 16325

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 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-ram 16327

This theorem is referenced by: ramub1 16354

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