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Theorem ramub1lem2 15674
Description: Lemma for ramub1 15675. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ramub1.m (𝜑𝑀 ∈ ℕ)
ramub1.r (𝜑𝑅 ∈ Fin)
ramub1.f (𝜑𝐹:𝑅⟶ℕ)
ramub1.g 𝐺 = (𝑥𝑅 ↦ (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦)))))
ramub1.1 (𝜑𝐺:𝑅⟶ℕ0)
ramub1.2 (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0)
ramub1.3 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
ramub1.4 (𝜑𝑆 ∈ Fin)
ramub1.5 (𝜑 → (#‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))
ramub1.6 (𝜑𝐾:(𝑆𝐶𝑀)⟶𝑅)
ramub1.x (𝜑𝑋𝑆)
ramub1.h 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))
Assertion
Ref Expression
ramub1lem2 (𝜑 → ∃𝑐𝑅𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘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.
StepHypRef Expression
1 ramub1.3 . . 3 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
2 ramub1.m . . . 4 (𝜑𝑀 ∈ ℕ)
3 nnm1nn0 11294 . . . 4 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
42, 3syl 17 . . 3 (𝜑 → (𝑀 − 1) ∈ ℕ0)
5 ramub1.r . . 3 (𝜑𝑅 ∈ Fin)
6 ramub1.1 . . 3 (𝜑𝐺:𝑅⟶ℕ0)
7 ramub1.2 . . 3 (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0)
8 ramub1.4 . . . 4 (𝜑𝑆 ∈ Fin)
9 diffi 8152 . . . 4 (𝑆 ∈ Fin → (𝑆 ∖ {𝑋}) ∈ Fin)
108, 9syl 17 . . 3 (𝜑 → (𝑆 ∖ {𝑋}) ∈ Fin)
117nn0red 11312 . . . . 5 (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℝ)
1211leidd 10554 . . . 4 (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ≤ ((𝑀 − 1) Ramsey 𝐺))
13 hashcl 13103 . . . . . . 7 ((𝑆 ∖ {𝑋}) ∈ Fin → (#‘(𝑆 ∖ {𝑋})) ∈ ℕ0)
1410, 13syl 17 . . . . . 6 (𝜑 → (#‘(𝑆 ∖ {𝑋})) ∈ ℕ0)
1514nn0cnd 11313 . . . . 5 (𝜑 → (#‘(𝑆 ∖ {𝑋})) ∈ ℂ)
167nn0cnd 11313 . . . . 5 (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℂ)
17 1cnd 10016 . . . . 5 (𝜑 → 1 ∈ ℂ)
18 undif1 4021 . . . . . . . 8 ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = (𝑆 ∪ {𝑋})
19 ramub1.x . . . . . . . . . 10 (𝜑𝑋𝑆)
2019snssd 4316 . . . . . . . . 9 (𝜑 → {𝑋} ⊆ 𝑆)
21 ssequn2 3770 . . . . . . . . 9 ({𝑋} ⊆ 𝑆 ↔ (𝑆 ∪ {𝑋}) = 𝑆)
2220, 21sylib 208 . . . . . . . 8 (𝜑 → (𝑆 ∪ {𝑋}) = 𝑆)
2318, 22syl5eq 2667 . . . . . . 7 (𝜑 → ((𝑆 ∖ {𝑋}) ∪ {𝑋}) = 𝑆)
2423fveq2d 6162 . . . . . 6 (𝜑 → (#‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = (#‘𝑆))
25 neldifsnd 4298 . . . . . . 7 (𝜑 → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
26 hashunsng 13137 . . . . . . . 8 (𝑋𝑆 → (((𝑆 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) → (#‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = ((#‘(𝑆 ∖ {𝑋})) + 1)))
2719, 26syl 17 . . . . . . 7 (𝜑 → (((𝑆 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑆 ∖ {𝑋})) → (#‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = ((#‘(𝑆 ∖ {𝑋})) + 1)))
2810, 25, 27mp2and 714 . . . . . 6 (𝜑 → (#‘((𝑆 ∖ {𝑋}) ∪ {𝑋})) = ((#‘(𝑆 ∖ {𝑋})) + 1))
29 ramub1.5 . . . . . 6 (𝜑 → (#‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))
3024, 28, 293eqtr3d 2663 . . . . 5 (𝜑 → ((#‘(𝑆 ∖ {𝑋})) + 1) = (((𝑀 − 1) Ramsey 𝐺) + 1))
3115, 16, 17, 30addcan2ad 10202 . . . 4 (𝜑 → (#‘(𝑆 ∖ {𝑋})) = ((𝑀 − 1) Ramsey 𝐺))
3212, 31breqtrrd 4651 . . 3 (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ≤ (#‘(𝑆 ∖ {𝑋})))
33 ramub1.6 . . . . . 6 (𝜑𝐾:(𝑆𝐶𝑀)⟶𝑅)
3433adantr 481 . . . . 5 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
351hashbcval 15649 . . . . . . . . . . . . . . 15 (((𝑆 ∖ {𝑋}) ∈ Fin ∧ (𝑀 − 1) ∈ ℕ0) → ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) = {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (#‘𝑥) = (𝑀 − 1)})
3610, 4, 35syl2anc 692 . . . . . . . . . . . . . 14 (𝜑 → ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) = {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (#‘𝑥) = (𝑀 − 1)})
3736eleq2d 2684 . . . . . . . . . . . . 13 (𝜑 → (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↔ 𝑢 ∈ {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (#‘𝑥) = (𝑀 − 1)}))
38 fveq2 6158 . . . . . . . . . . . . . . 15 (𝑥 = 𝑢 → (#‘𝑥) = (#‘𝑢))
3938eqeq1d 2623 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → ((#‘𝑥) = (𝑀 − 1) ↔ (#‘𝑢) = (𝑀 − 1)))
4039elrab 3351 . . . . . . . . . . . . 13 (𝑢 ∈ {𝑥 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∣ (#‘𝑥) = (𝑀 − 1)} ↔ (𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∧ (#‘𝑢) = (𝑀 − 1)))
4137, 40syl6bb 276 . . . . . . . . . . . 12 (𝜑 → (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↔ (𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}) ∧ (#‘𝑢) = (𝑀 − 1))))
4241simprbda 652 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ∈ 𝒫 (𝑆 ∖ {𝑋}))
4342elpwid 4148 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ⊆ (𝑆 ∖ {𝑋}))
4443difss2d 3724 . . . . . . . . 9 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢𝑆)
4520adantr 481 . . . . . . . . 9 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → {𝑋} ⊆ 𝑆)
4644, 45unssd 3773 . . . . . . . 8 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ⊆ 𝑆)
47 vex 3193 . . . . . . . . . 10 𝑢 ∈ V
48 snex 4879 . . . . . . . . . 10 {𝑋} ∈ V
4947, 48unex 6921 . . . . . . . . 9 (𝑢 ∪ {𝑋}) ∈ V
5049elpw 4142 . . . . . . . 8 ((𝑢 ∪ {𝑋}) ∈ 𝒫 𝑆 ↔ (𝑢 ∪ {𝑋}) ⊆ 𝑆)
5146, 50sylibr 224 . . . . . . 7 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ 𝒫 𝑆)
5210adantr 481 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑆 ∖ {𝑋}) ∈ Fin)
53 ssfi 8140 . . . . . . . . . 10 (((𝑆 ∖ {𝑋}) ∈ Fin ∧ 𝑢 ⊆ (𝑆 ∖ {𝑋})) → 𝑢 ∈ Fin)
5452, 43, 53syl2anc 692 . . . . . . . . 9 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑢 ∈ Fin)
55 neldifsnd 4298 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ¬ 𝑋 ∈ (𝑆 ∖ {𝑋}))
5643, 55ssneldd 3591 . . . . . . . . 9 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ¬ 𝑋𝑢)
5719adantr 481 . . . . . . . . . 10 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → 𝑋𝑆)
58 hashunsng 13137 . . . . . . . . . 10 (𝑋𝑆 → ((𝑢 ∈ Fin ∧ ¬ 𝑋𝑢) → (#‘(𝑢 ∪ {𝑋})) = ((#‘𝑢) + 1)))
5957, 58syl 17 . . . . . . . . 9 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((𝑢 ∈ Fin ∧ ¬ 𝑋𝑢) → (#‘(𝑢 ∪ {𝑋})) = ((#‘𝑢) + 1)))
6054, 56, 59mp2and 714 . . . . . . . 8 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (#‘(𝑢 ∪ {𝑋})) = ((#‘𝑢) + 1))
6141simplbda 653 . . . . . . . . 9 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (#‘𝑢) = (𝑀 − 1))
6261oveq1d 6630 . . . . . . . 8 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((#‘𝑢) + 1) = ((𝑀 − 1) + 1))
632nncnd 10996 . . . . . . . . . 10 (𝜑𝑀 ∈ ℂ)
64 ax-1cn 9954 . . . . . . . . . 10 1 ∈ ℂ
65 npcan 10250 . . . . . . . . . 10 ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀)
6663, 64, 65sylancl 693 . . . . . . . . 9 (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
6766adantr 481 . . . . . . . 8 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
6860, 62, 673eqtrd 2659 . . . . . . 7 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (#‘(𝑢 ∪ {𝑋})) = 𝑀)
69 fveq2 6158 . . . . . . . . 9 (𝑥 = (𝑢 ∪ {𝑋}) → (#‘𝑥) = (#‘(𝑢 ∪ {𝑋})))
7069eqeq1d 2623 . . . . . . . 8 (𝑥 = (𝑢 ∪ {𝑋}) → ((#‘𝑥) = 𝑀 ↔ (#‘(𝑢 ∪ {𝑋})) = 𝑀))
7170elrab 3351 . . . . . . 7 ((𝑢 ∪ {𝑋}) ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (#‘𝑥) = 𝑀} ↔ ((𝑢 ∪ {𝑋}) ∈ 𝒫 𝑆 ∧ (#‘(𝑢 ∪ {𝑋})) = 𝑀))
7251, 68, 71sylanbrc 697 . . . . . 6 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (#‘𝑥) = 𝑀})
732nnnn0d 11311 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
741hashbcval 15649 . . . . . . . 8 ((𝑆 ∈ Fin ∧ 𝑀 ∈ ℕ0) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (#‘𝑥) = 𝑀})
758, 73, 74syl2anc 692 . . . . . . 7 (𝜑 → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (#‘𝑥) = 𝑀})
7675adantr 481 . . . . . 6 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (#‘𝑥) = 𝑀})
7772, 76eleqtrrd 2701 . . . . 5 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝑢 ∪ {𝑋}) ∈ (𝑆𝐶𝑀))
7834, 77ffvelrnd 6326 . . . 4 ((𝜑𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))) → (𝐾‘(𝑢 ∪ {𝑋})) ∈ 𝑅)
79 ramub1.h . . . 4 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))
8078, 79fmptd 6351 . . 3 (𝜑𝐻:((𝑆 ∖ {𝑋})𝐶(𝑀 − 1))⟶𝑅)
811, 4, 5, 6, 7, 10, 32, 80rami 15662 . 2 (𝜑 → ∃𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))
8273adantr 481 . . . . . 6 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → 𝑀 ∈ ℕ0)
835adantr 481 . . . . . 6 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → 𝑅 ∈ Fin)
84 ramub1.f . . . . . . . . . . . 12 (𝜑𝐹:𝑅⟶ℕ)
8584adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → 𝐹:𝑅⟶ℕ)
86 simprll 801 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → 𝑑𝑅)
8785, 86ffvelrnd 6326 . . . . . . . . . 10 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → (𝐹𝑑) ∈ ℕ)
88 nnm1nn0 11294 . . . . . . . . . 10 ((𝐹𝑑) ∈ ℕ → ((𝐹𝑑) − 1) ∈ ℕ0)
8987, 88syl 17 . . . . . . . . 9 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → ((𝐹𝑑) − 1) ∈ ℕ0)
9089adantr 481 . . . . . . . 8 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ 𝑦𝑅) → ((𝐹𝑑) − 1) ∈ ℕ0)
9185ffvelrnda 6325 . . . . . . . . 9 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ 𝑦𝑅) → (𝐹𝑦) ∈ ℕ)
9291nnnn0d 11311 . . . . . . . 8 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ 𝑦𝑅) → (𝐹𝑦) ∈ ℕ0)
9390, 92ifcld 4109 . . . . . . 7 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ 𝑦𝑅) → if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)) ∈ ℕ0)
94 eqid 2621 . . . . . . 7 (𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦))) = (𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))
9593, 94fmptd 6351 . . . . . 6 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → (𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦))):𝑅⟶ℕ0)
96 equequ2 1950 . . . . . . . . . . . 12 (𝑥 = 𝑑 → (𝑦 = 𝑥𝑦 = 𝑑))
97 fveq2 6158 . . . . . . . . . . . . 13 (𝑥 = 𝑑 → (𝐹𝑥) = (𝐹𝑑))
9897oveq1d 6630 . . . . . . . . . . . 12 (𝑥 = 𝑑 → ((𝐹𝑥) − 1) = ((𝐹𝑑) − 1))
9996, 98ifbieq1d 4087 . . . . . . . . . . 11 (𝑥 = 𝑑 → if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦)) = if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))
10099mpteq2dv 4715 . . . . . . . . . 10 (𝑥 = 𝑑 → (𝑦𝑅 ↦ if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦))) = (𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦))))
101100oveq2d 6631 . . . . . . . . 9 (𝑥 = 𝑑 → (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦)))) = (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))))
102 ramub1.g . . . . . . . . 9 𝐺 = (𝑥𝑅 ↦ (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦)))))
103 ovex 6643 . . . . . . . . 9 (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))) ∈ V
104101, 102, 103fvmpt 6249 . . . . . . . 8 (𝑑𝑅 → (𝐺𝑑) = (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))))
10586, 104syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → (𝐺𝑑) = (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))))
1066adantr 481 . . . . . . . 8 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → 𝐺:𝑅⟶ℕ0)
107106, 86ffvelrnd 6326 . . . . . . 7 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → (𝐺𝑑) ∈ ℕ0)
108105, 107eqeltrrd 2699 . . . . . 6 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))) ∈ ℕ0)
109 simprlr 802 . . . . . 6 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → 𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋}))
110 simprrl 803 . . . . . . 7 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → (𝐺𝑑) ≤ (#‘𝑤))
111105, 110eqbrtrrd 4647 . . . . . 6 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))) ≤ (#‘𝑤))
11233adantr 481 . . . . . . 7 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
1138adantr 481 . . . . . . . 8 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → 𝑆 ∈ Fin)
114109elpwid 4148 . . . . . . . . 9 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → 𝑤 ⊆ (𝑆 ∖ {𝑋}))
115114difss2d 3724 . . . . . . . 8 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → 𝑤𝑆)
1161hashbcss 15651 . . . . . . . 8 ((𝑆 ∈ Fin ∧ 𝑤𝑆𝑀 ∈ ℕ0) → (𝑤𝐶𝑀) ⊆ (𝑆𝐶𝑀))
117113, 115, 82, 116syl3anc 1323 . . . . . . 7 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → (𝑤𝐶𝑀) ⊆ (𝑆𝐶𝑀))
118112, 117fssresd 6038 . . . . . 6 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → (𝐾 ↾ (𝑤𝐶𝑀)):(𝑤𝐶𝑀)⟶𝑅)
1191, 82, 83, 95, 108, 109, 111, 118rami 15662 . . . . 5 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → ∃𝑐𝑅𝑣 ∈ 𝒫 𝑤(((𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))‘𝑐) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))
120 equequ1 1949 . . . . . . . . . . . . . 14 (𝑦 = 𝑐 → (𝑦 = 𝑑𝑐 = 𝑑))
121 fveq2 6158 . . . . . . . . . . . . . 14 (𝑦 = 𝑐 → (𝐹𝑦) = (𝐹𝑐))
122120, 121ifbieq2d 4089 . . . . . . . . . . . . 13 (𝑦 = 𝑐 → if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)) = if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)))
123 ovex 6643 . . . . . . . . . . . . . 14 ((𝐹𝑑) − 1) ∈ V
124 fvex 6168 . . . . . . . . . . . . . 14 (𝐹𝑐) ∈ V
125123, 124ifex 4134 . . . . . . . . . . . . 13 if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ∈ V
126122, 94, 125fvmpt 6249 . . . . . . . . . . . 12 (𝑐𝑅 → ((𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))‘𝑐) = if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)))
127126ad2antrl 763 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ (𝑐𝑅𝑣 ∈ 𝒫 𝑤)) → ((𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))‘𝑐) = if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)))
128127breq1d 4633 . . . . . . . . . 10 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ (𝑐𝑅𝑣 ∈ 𝒫 𝑤)) → (((𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))‘𝑐) ≤ (#‘𝑣) ↔ if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣)))
129128anbi1d 740 . . . . . . . . 9 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ (𝑐𝑅𝑣 ∈ 𝒫 𝑤)) → ((((𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))‘𝑐) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) ↔ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}))))
1302ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑀 ∈ ℕ)
1315ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑅 ∈ Fin)
13284ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝐹:𝑅⟶ℕ)
1336ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝐺:𝑅⟶ℕ0)
1347ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0)
1358ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑆 ∈ Fin)
13629ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (#‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))
13733ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝐾:(𝑆𝐶𝑀)⟶𝑅)
13819ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑋𝑆)
13986adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑑𝑅)
140114adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑤 ⊆ (𝑆 ∖ {𝑋}))
141110adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝐺𝑑) ≤ (#‘𝑤))
142 simprrr 804 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑}))
143142adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑}))
144 simprll 801 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑐𝑅)
145 simprlr 802 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑣 ∈ 𝒫 𝑤)
146145elpwid 4148 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → 𝑣𝑤)
147 simprrl 803 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣))
148 simprrr 804 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}))
149 cnvresima 5592 . . . . . . . . . . . . 13 ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}) = ((𝐾 “ {𝑐}) ∩ (𝑤𝐶𝑀))
150 inss1 3817 . . . . . . . . . . . . 13 ((𝐾 “ {𝑐}) ∩ (𝑤𝐶𝑀)) ⊆ (𝐾 “ {𝑐})
151149, 150eqsstri 3620 . . . . . . . . . . . 12 ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐}) ⊆ (𝐾 “ {𝑐})
152148, 151syl6ss 3600 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → (𝑣𝐶𝑀) ⊆ (𝐾 “ {𝑐}))
153130, 131, 132, 102, 133, 134, 1, 135, 136, 137, 138, 79, 139, 140, 141, 143, 144, 146, 147, 152ramub1lem1 15673 . . . . . . . . . 10 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ ((𝑐𝑅𝑣 ∈ 𝒫 𝑤) ∧ (if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})))) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐})))
154153expr 642 . . . . . . . . 9 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ (𝑐𝑅𝑣 ∈ 𝒫 𝑤)) → ((if(𝑐 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑐)) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐}))))
155129, 154sylbid 230 . . . . . . . 8 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ (𝑐𝑅𝑣 ∈ 𝒫 𝑤)) → ((((𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))‘𝑐) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐}))))
156155anassrs 679 . . . . . . 7 ((((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ 𝑐𝑅) ∧ 𝑣 ∈ 𝒫 𝑤) → ((((𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))‘𝑐) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐}))))
157156rexlimdva 3026 . . . . . 6 (((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) ∧ 𝑐𝑅) → (∃𝑣 ∈ 𝒫 𝑤(((𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))‘𝑐) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐}))))
158157reximdva 3013 . . . . 5 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → (∃𝑐𝑅𝑣 ∈ 𝒫 𝑤(((𝑦𝑅 ↦ if(𝑦 = 𝑑, ((𝐹𝑑) − 1), (𝐹𝑦)))‘𝑐) ≤ (#‘𝑣) ∧ (𝑣𝐶𝑀) ⊆ ((𝐾 ↾ (𝑤𝐶𝑀)) “ {𝑐})) → ∃𝑐𝑅𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐}))))
159119, 158mpd 15 . . . 4 ((𝜑 ∧ ((𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})) ∧ ((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})))) → ∃𝑐𝑅𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐})))
160159expr 642 . . 3 ((𝜑 ∧ (𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋}))) → (((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})) → ∃𝑐𝑅𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐}))))
161160rexlimdvva 3033 . 2 (𝜑 → (∃𝑑𝑅𝑤 ∈ 𝒫 (𝑆 ∖ {𝑋})((𝐺𝑑) ≤ (#‘𝑤) ∧ (𝑤𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝑑})) → ∃𝑐𝑅𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐}))))
16281, 161mpd 15 1 (𝜑 → ∃𝑐𝑅𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2909  {crab 2912  Vcvv 3190  cdif 3557  cun 3558  cin 3559  wss 3560  ifcif 4064  𝒫 cpw 4136  {csn 4155   class class class wbr 4623  cmpt 4683  ccnv 5083  cres 5086  cima 5087  wf 5853  cfv 5857  (class class class)co 6615  cmpt2 6617  Fincfn 7915  cc 9894  1c1 9897   + caddc 9899  cle 10035  cmin 10226  cn 10980  0cn0 11252  #chash 13073   Ramsey cram 15646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-hash 13074  df-ram 15648
This theorem is referenced by:  ramub1  15675
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