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Theorem ramub1lem2 17047

Description: Lemma for ramub1 17048. (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 12542	. . . 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 9189	. . . 4 ⊢ (𝑆 ∈ Fin → (𝑆 ∖ {𝑋}) ∈ Fin)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → (𝑆 ∖ {𝑋}) ∈ Fin)
11	7	nn0red 12563	. . . . 5 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℝ)
12	11	leidd 11803	. . . 4 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ≤ ((𝑀 − 1) Ramsey 𝐺))
13		hashcl 14374	. . . . . . 7 ⊢ ((𝑆 ∖ {𝑋}) ∈ Fin → (♯‘(𝑆 ∖ {𝑋})) ∈ ℕ₀)
14	10, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) ∈ ℕ₀)
15	14	nn0cnd 12564	. . . . 5 ⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) ∈ ℂ)
16	7	nn0cnd 12564	. . . . 5 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℂ)
17		1cnd 11230	. . . . 5 ⊢ (𝜑 → 1 ∈ ℂ)
18		undif1 4451	. . . . . . . 8 ⊢ ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = (𝑆 ∪ {𝑋})
19		ramub1.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
20	19	snssd 4785	. . . . . . . . 9 ⊢ (𝜑 → {𝑋} ⊆ 𝑆)
21		ssequn2 4164	. . . . . . . . 9 ⊢ ({𝑋} ⊆ 𝑆 ↔ (𝑆 ∪ {𝑋}) = 𝑆)
22	20, 21	sylib 218	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ {𝑋}) = 𝑆)
23	18, 22	eqtrid 2782	. . . . . . 7 ⊢ (𝜑 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
24	23	fveq2d 6880	. . . . . 6 ⊢ (𝜑 → (♯‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = (♯‘𝑆))
25		neldifsnd 4769	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
26		hashunsng 14410	. . . . . . . 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 699	. . . . . 6 ⊢ (𝜑 → (♯‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑆 ∖ {𝑋})) + 1))
29		ramub1.5	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))
30	24, 28, 29	3eqtr3d 2778	. . . . 5 ⊢ (𝜑 → ((♯‘(𝑆 ∖ {𝑋})) + 1) = (((𝑀 − 1) Ramsey 𝐺) + 1))
31	15, 16, 17, 30	addcan2ad 11441	. . . 4 ⊢ (𝜑 → (♯‘(𝑆 ∖ {𝑋})) = ((𝑀 − 1) Ramsey 𝐺))
32	12, 31	breqtrrd 5147	. . 3 ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ≤ (♯‘(𝑆 ∖ {𝑋})))
33		ramub1.6	. . . . . 6 ⊢ (𝜑 → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
34	33	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
35		fveqeq2 6885	. . . . . . 7 ⊢ (𝑥 = (𝑢 ∪ {𝑋}) → ((♯‘𝑥) = 𝑀 ↔ (♯‘(𝑢 ∪ {𝑋})) = 𝑀))
36	1	hashbcval 17022	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∖ {𝑋}) ∈ Fin ∧ (𝑀 − 1) ∈ ℕ₀) → ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) = {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)})
37	10, 4, 36	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) = {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)})
38	37	eleq2d 2820	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↔ 𝑢 ∈ {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)}))
39		fveqeq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → ((♯‘𝑥) = (𝑀 − 1) ↔ (♯‘𝑢) = (𝑀 − 1)))
40	39	elrab 3671	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (♯‘𝑥) = (𝑀 − 1)} ↔ (𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∧ (♯‘𝑢) = (𝑀 − 1)))
41	38, 40	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↔ (𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∧ (♯‘𝑢) = (𝑀 − 1))))
42	41	simprbda 498	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}))
43	42	elpwid 4584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ⊆ (𝑆 ∖ {𝑋}))
44	43	difss2d 4114	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ⊆ 𝑆)
45	20	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → {𝑋} ⊆ 𝑆)
46	44, 45	unssd 4167	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ⊆ 𝑆)
47		vex 3463	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
48		snex 5406	. . . . . . . . . 10 ⊢ {𝑋} ∈ V
49	47, 48	unex 7738	. . . . . . . . 9 ⊢ (𝑢 ∪ {𝑋}) ∈ V
50	49	elpw 4579	. . . . . . . 8 ⊢ ((𝑢 ∪ {𝑋}) ∈ 𝒫 𝑆 ↔ (𝑢 ∪ {𝑋}) ⊆ 𝑆)
51	46, 50	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ 𝒫 𝑆)
52	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑆 ∖ {𝑋}) ∈ Fin)
53	52, 43	ssfid 9273	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ∈ Fin)
54		neldifsnd 4769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
55	43, 54	ssneldd 3961	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ¬ 𝑋 ∈ 𝑢)
56	19	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑋 ∈ 𝑆)
57		hashunsng 14410	. . . . . . . . . 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 699	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (♯‘(𝑢 ∪ {𝑋})) = ((♯‘𝑢) + 1))
60	41	simplbda 499	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (♯‘𝑢) = (𝑀 − 1))
61	60	oveq1d 7420	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((♯‘𝑢) + 1) = ((𝑀 − 1) + 1))
62	2	nncnd 12256	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
63		ax-1cn 11187	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
64		npcan 11491	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀)
65	62, 63, 64	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
66	65	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
67	59, 61, 66	3eqtrd 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (♯‘(𝑢 ∪ {𝑋})) = 𝑀)
68	35, 51, 67	elrabd 3673	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
69	2	nnnn0d 12562	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
70	1	hashbcval 17022	. . . . . . . 8 ⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ ℕ₀) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
71	8, 69, 70	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
72	71	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
73	68, 72	eleqtrrd 2837	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ (𝑆𝐶𝑀))
74	34, 73	ffvelcdmd 7075	. . . 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 7104	. . 3 ⊢ (𝜑 → 𝐻:((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))⟶𝑅)
77	1, 4, 5, 6, 7, 10, 32, 76	rami 17035	. 2 ⊢ (𝜑 → ∃𝑑 ∈ 𝑅 ∃𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))
78	69	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑀 ∈ ℕ₀)
79	5	adantr 480	. . . . . 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 480	. . . . . . . . . . 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	ffvelcdmd 7075	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝐹‘𝑑) ∈ ℕ)
84		nnm1nn0 12542	. . . . . . . . . 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 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ 𝑦 ∈ 𝑅) → ((𝐹‘𝑑) − 1) ∈ ℕ₀)
87	81	ffvelcdmda 7074	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) ∧ 𝑦 ∈ 𝑅) → (𝐹‘𝑦) ∈ ℕ)
88	87	nnnn0d 12562	. . . . . . . 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 4547	. . . . . . 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 2735	. . . . . . 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 7104	. . . . . 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 2025	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑑 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑑))
93		fveq2 6876	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑑 → (𝐹‘𝑥) = (𝐹‘𝑑))
94	93	oveq1d 7420	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑑 → ((𝐹‘𝑥) − 1) = ((𝐹‘𝑑) − 1))
95	92, 94	ifbieq1d 4525	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑑 → if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦)) = if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))
96	95	mpteq2dv 5215	. . . . . . . . . 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 7421	. . . . . . . . 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 7438	. . . . . . . . 9 ⊢ (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)))) ∈ V
100	97, 98, 99	fvmpt 6986	. . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝐺:𝑅⟶ℕ₀)
103	102, 82	ffvelcdmd 7075	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝐺‘𝑑) ∈ ℕ₀)
104	101, 103	eqeltrrd 2835	. . . . . 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 5143	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
109	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑆 ∈ Fin)
110	105	elpwid 4584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑤 ⊆ (𝑆 ∖ {𝑋}))
111	110	difss2d 4114	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → 𝑤 ⊆ 𝑆)
112	1	hashbcss 17024	. . . . . . . 8 ⊢ ((𝑆 ∈ Fin ∧ 𝑤 ⊆ 𝑆 ∧ 𝑀 ∈ ℕ₀) → (𝑤𝐶𝑀) ⊆ (𝑆𝐶𝑀))
113	109, 111, 78, 112	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑑 ∈ 𝑅 ∧ 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺‘𝑑) ≤ (♯‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (^◡𝐻 “ {𝑑})))) → (𝑤𝐶𝑀) ⊆ (𝑆𝐶𝑀))
114	108, 113	fssresd 6745	. . . . . 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 17035	. . . . 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 2024	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑐 → (𝑦 = 𝑑 ↔ 𝑐 = 𝑑))
117		fveq2 6876	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑐 → (𝐹‘𝑦) = (𝐹‘𝑐))
118	116, 117	ifbieq2d 4527	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑐 → if(𝑦 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑦)) = if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)))
119		ovex 7438	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑑) − 1) ∈ V
120		fvex 6889	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑐) ∈ V
121	119, 120	ifex 4551	. . . . . . . . . . . . 13 ⊢ if(𝑐 = 𝑑, ((𝐹‘𝑑) − 1), (𝐹‘𝑐)) ∈ V
122	118, 90, 121	fvmpt 6986	. . . . . . . . . . . 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 728	. . . . . . . . . . 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 5129	. . . . . . . . . 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 631	. . . . . . . . 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 726	. . . . . . . . . . 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 726	. . . . . . . . . . 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 726	. . . . . . . . . . 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 726	. . . . . . . . . . 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 726	. . . . . . . . . . 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 726	. . . . . . . . . . 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 726	. . . . . . . . . . 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 726	. . . . . . . . . . 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 726	. . . . . . . . . . 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 480	. . . . . . . . . . 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 480	. . . . . . . . . . 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 480	. . . . . . . . . . 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 480	. . . . . . . . . . 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 4584	. . . . . . . . . . 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 6219	. . . . . . . . . . . . 13 ⊢ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}) = ((^◡𝐾 “ {𝑐}) ∩ (𝑤𝐶𝑀))
146		inss1 4212	. . . . . . . . . . . . 13 ⊢ ((^◡𝐾 “ {𝑐}) ∩ (𝑤𝐶𝑀)) ⊆ (^◡𝐾 “ {𝑐})
147	145, 146	eqsstri 4005	. . . . . . . . . . . 12 ⊢ (^◡(𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}) ⊆ (^◡𝐾 “ {𝑐})
148	144, 147	sstrdi 3971	. . . . . . . . . . 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 17046	. . . . . . . . . 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 456	. . . . . . . . 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 240	. . . . . . . 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 467	. . . . . . 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 3141	. . . . . 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 3153	. . . . 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 456	. . 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 3198	. 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 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3060 {crab 3415 Vcvv 3459 ∖ cdif 3923 ∪ cun 3924 ∩ cin 3925 ⊆ wss 3926 ifcif 4500 𝒫 cpw 4575 {csn 4601 class class class wbr 5119 ↦ cmpt 5201 ^◡ccnv 5653 ↾ cres 5656 “ cima 5657 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ∈ cmpo 7407 Fincfn 8959 ℂcc 11127 1c1 11130 + caddc 11132 ≤ cle 11270 − cmin 11466 ℕcn 12240 ℕ₀cn0 12501 ♯chash 14348 Ramsey cram 17019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-inf 9455 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-hash 14349 df-ram 17021

This theorem is referenced by: ramub1 17048

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