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Theorem ramub1lem1 15938
Description: Lemma for ramub1 15940. (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})))
ramub1.d (𝜑𝐷𝑅)
ramub1.w (𝜑𝑊 ⊆ (𝑆 ∖ {𝑋}))
ramub1.7 (𝜑 → (𝐺𝐷) ≤ (♯‘𝑊))
ramub1.8 (𝜑 → (𝑊𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝐷}))
ramub1.e (𝜑𝐸𝑅)
ramub1.v (𝜑𝑉𝑊)
ramub1.9 (𝜑 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (♯‘𝑉))
ramub1.s (𝜑 → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
Assertion
Ref Expression
ramub1lem1 (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
Distinct variable groups:   𝑥,𝑢,𝐷   𝑦,𝑢,𝑧,𝐹,𝑥   𝑎,𝑏,𝑖,𝑢,𝑥,𝑦,𝑧,𝑀   𝐺,𝑎,𝑖,𝑢,𝑥,𝑦,𝑧   𝑢,𝑅,𝑥,𝑦,𝑧   𝑊,𝑎,𝑖,𝑢   𝜑,𝑢,𝑥,𝑦,𝑧   𝑆,𝑎,𝑖,𝑢,𝑥,𝑦,𝑧   𝑉,𝑎,𝑖,𝑥,𝑧   𝑢,𝐶,𝑥,𝑦,𝑧   𝑢,𝐻,𝑥,𝑦,𝑧   𝑢,𝐾,𝑥,𝑦,𝑧   𝑥,𝐸,𝑧   𝑋,𝑎,𝑖,𝑢,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑖,𝑎,𝑏)   𝐶(𝑖,𝑎,𝑏)   𝐷(𝑦,𝑧,𝑖,𝑎,𝑏)   𝑅(𝑖,𝑎,𝑏)   𝑆(𝑏)   𝐸(𝑦,𝑢,𝑖,𝑎,𝑏)   𝐹(𝑖,𝑎,𝑏)   𝐺(𝑏)   𝐻(𝑖,𝑎,𝑏)   𝐾(𝑖,𝑎,𝑏)   𝑉(𝑦,𝑢,𝑏)   𝑊(𝑥,𝑦,𝑧,𝑏)   𝑋(𝑏)

Proof of Theorem ramub1lem1
StepHypRef Expression
1 ramub1.v . . . . . . . 8 (𝜑𝑉𝑊)
2 ramub1.w . . . . . . . 8 (𝜑𝑊 ⊆ (𝑆 ∖ {𝑋}))
31, 2sstrd 3763 . . . . . . 7 (𝜑𝑉 ⊆ (𝑆 ∖ {𝑋}))
43difss2d 3892 . . . . . 6 (𝜑𝑉𝑆)
5 ramub1.x . . . . . . 7 (𝜑𝑋𝑆)
65snssd 4476 . . . . . 6 (𝜑 → {𝑋} ⊆ 𝑆)
74, 6unssd 3941 . . . . 5 (𝜑 → (𝑉 ∪ {𝑋}) ⊆ 𝑆)
8 ramub1.4 . . . . . 6 (𝜑𝑆 ∈ Fin)
9 elpw2g 4959 . . . . . 6 (𝑆 ∈ Fin → ((𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆 ↔ (𝑉 ∪ {𝑋}) ⊆ 𝑆))
108, 9syl 17 . . . . 5 (𝜑 → ((𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆 ↔ (𝑉 ∪ {𝑋}) ⊆ 𝑆))
117, 10mpbird 247 . . . 4 (𝜑 → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
1211adantr 466 . . 3 ((𝜑𝐸 = 𝐷) → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
13 iftrue 4232 . . . . . . 7 (𝐸 = 𝐷 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = ((𝐹𝐷) − 1))
1413adantl 467 . . . . . 6 ((𝜑𝐸 = 𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = ((𝐹𝐷) − 1))
15 ramub1.9 . . . . . . 7 (𝜑 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (♯‘𝑉))
1615adantr 466 . . . . . 6 ((𝜑𝐸 = 𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (♯‘𝑉))
1714, 16eqbrtrrd 4811 . . . . 5 ((𝜑𝐸 = 𝐷) → ((𝐹𝐷) − 1) ≤ (♯‘𝑉))
18 ramub1.f . . . . . . . . 9 (𝜑𝐹:𝑅⟶ℕ)
19 ramub1.d . . . . . . . . 9 (𝜑𝐷𝑅)
2018, 19ffvelrnd 6504 . . . . . . . 8 (𝜑 → (𝐹𝐷) ∈ ℕ)
2120adantr 466 . . . . . . 7 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ∈ ℕ)
2221nnred 11238 . . . . . 6 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ∈ ℝ)
23 1red 10258 . . . . . 6 ((𝜑𝐸 = 𝐷) → 1 ∈ ℝ)
24 ssfi 8337 . . . . . . . . 9 ((𝑆 ∈ Fin ∧ 𝑉𝑆) → 𝑉 ∈ Fin)
258, 4, 24syl2anc 567 . . . . . . . 8 (𝜑𝑉 ∈ Fin)
26 hashcl 13350 . . . . . . . 8 (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
27 nn0re 11504 . . . . . . . 8 ((♯‘𝑉) ∈ ℕ0 → (♯‘𝑉) ∈ ℝ)
2825, 26, 273syl 18 . . . . . . 7 (𝜑 → (♯‘𝑉) ∈ ℝ)
2928adantr 466 . . . . . 6 ((𝜑𝐸 = 𝐷) → (♯‘𝑉) ∈ ℝ)
3022, 23, 29lesubaddd 10827 . . . . 5 ((𝜑𝐸 = 𝐷) → (((𝐹𝐷) − 1) ≤ (♯‘𝑉) ↔ (𝐹𝐷) ≤ ((♯‘𝑉) + 1)))
3117, 30mpbid 222 . . . 4 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ≤ ((♯‘𝑉) + 1))
32 fveq2 6333 . . . . 5 (𝐸 = 𝐷 → (𝐹𝐸) = (𝐹𝐷))
33 snidg 4346 . . . . . . . 8 (𝑋𝑆𝑋 ∈ {𝑋})
345, 33syl 17 . . . . . . 7 (𝜑𝑋 ∈ {𝑋})
353sseld 3752 . . . . . . . 8 (𝜑 → (𝑋𝑉𝑋 ∈ (𝑆 ∖ {𝑋})))
36 eldifn 3885 . . . . . . . 8 (𝑋 ∈ (𝑆 ∖ {𝑋}) → ¬ 𝑋 ∈ {𝑋})
3735, 36syl6 35 . . . . . . 7 (𝜑 → (𝑋𝑉 → ¬ 𝑋 ∈ {𝑋}))
3834, 37mt2d 133 . . . . . 6 (𝜑 → ¬ 𝑋𝑉)
39 hashunsng 13384 . . . . . . 7 (𝑋𝑆 → ((𝑉 ∈ Fin ∧ ¬ 𝑋𝑉) → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1)))
405, 39syl 17 . . . . . 6 (𝜑 → ((𝑉 ∈ Fin ∧ ¬ 𝑋𝑉) → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1)))
4125, 38, 40mp2and 673 . . . . 5 (𝜑 → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1))
4232, 41breqan12rd 4804 . . . 4 ((𝜑𝐸 = 𝐷) → ((𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ↔ (𝐹𝐷) ≤ ((♯‘𝑉) + 1)))
4331, 42mpbird 247 . . 3 ((𝜑𝐸 = 𝐷) → (𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})))
44 snfi 8195 . . . . . . 7 {𝑋} ∈ Fin
45 unfi 8384 . . . . . . 7 ((𝑉 ∈ Fin ∧ {𝑋} ∈ Fin) → (𝑉 ∪ {𝑋}) ∈ Fin)
4625, 44, 45sylancl 568 . . . . . 6 (𝜑 → (𝑉 ∪ {𝑋}) ∈ Fin)
47 ramub1.m . . . . . . 7 (𝜑𝑀 ∈ ℕ)
4847nnnn0d 11554 . . . . . 6 (𝜑𝑀 ∈ ℕ0)
49 ramub1.3 . . . . . . 7 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
5049hashbcval 15914 . . . . . 6 (((𝑉 ∪ {𝑋}) ∈ Fin ∧ 𝑀 ∈ ℕ0) → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
5146, 48, 50syl2anc 567 . . . . 5 (𝜑 → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
5251adantr 466 . . . 4 ((𝜑𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
53 simpl1l 1278 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝜑)
5449hashbcval 15914 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑀 ∈ ℕ0) → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
5525, 48, 54syl2anc 567 . . . . . . . . 9 (𝜑 → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
56 ramub1.s . . . . . . . . 9 (𝜑 → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
5755, 56eqsstr3d 3790 . . . . . . . 8 (𝜑 → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀} ⊆ (𝐾 “ {𝐸}))
5853, 57syl 17 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀} ⊆ (𝐾 “ {𝐸}))
59 simpr 471 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 𝑉)
60 simpl3 1231 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
61 rabid 3264 . . . . . . . 8 (𝑥 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀} ↔ (𝑥 ∈ 𝒫 𝑉 ∧ (♯‘𝑥) = 𝑀))
6259, 60, 61sylanbrc 566 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
6358, 62sseldd 3754 . . . . . 6 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝐾 “ {𝐸}))
64 simpl2 1229 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}))
6564elpwid 4310 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ (𝑉 ∪ {𝑋}))
66 simpl1l 1278 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝜑)
6766, 7syl 17 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑉 ∪ {𝑋}) ⊆ 𝑆)
6865, 67sstrd 3763 . . . . . . . . . 10 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥𝑆)
69 vex 3354 . . . . . . . . . . 11 𝑥 ∈ V
7069elpw 4304 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝑆𝑥𝑆)
7168, 70sylibr 224 . . . . . . . . 9 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 𝑆)
72 simpl3 1231 . . . . . . . . 9 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = 𝑀)
73 rabid 3264 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀} ↔ (𝑥 ∈ 𝒫 𝑆 ∧ (♯‘𝑥) = 𝑀))
7471, 72, 73sylanbrc 566 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
7549hashbcval 15914 . . . . . . . . . 10 ((𝑆 ∈ Fin ∧ 𝑀 ∈ ℕ0) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
768, 48, 75syl2anc 567 . . . . . . . . 9 (𝜑 → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
7766, 76syl 17 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
7874, 77eleqtrrd 2853 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝑆𝐶𝑀))
792difss2d 3892 . . . . . . . . . . . . . . 15 (𝜑𝑊𝑆)
80 ssfi 8337 . . . . . . . . . . . . . . 15 ((𝑆 ∈ Fin ∧ 𝑊𝑆) → 𝑊 ∈ Fin)
818, 79, 80syl2anc 567 . . . . . . . . . . . . . 14 (𝜑𝑊 ∈ Fin)
82 nnm1nn0 11537 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
8347, 82syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑀 − 1) ∈ ℕ0)
8449hashbcval 15914 . . . . . . . . . . . . . 14 ((𝑊 ∈ Fin ∧ (𝑀 − 1) ∈ ℕ0) → (𝑊𝐶(𝑀 − 1)) = {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
8581, 83, 84syl2anc 567 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐶(𝑀 − 1)) = {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
86 ramub1.8 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝐷}))
8785, 86eqsstr3d 3790 . . . . . . . . . . . 12 (𝜑 → {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)} ⊆ (𝐻 “ {𝐷}))
8866, 87syl 17 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)} ⊆ (𝐻 “ {𝐷}))
89 uncom 3909 . . . . . . . . . . . . . . . 16 (𝑉 ∪ {𝑋}) = ({𝑋} ∪ 𝑉)
9065, 89syl6sseq 3801 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ ({𝑋} ∪ 𝑉))
91 ssundif 4195 . . . . . . . . . . . . . . 15 (𝑥 ⊆ ({𝑋} ∪ 𝑉) ↔ (𝑥 ∖ {𝑋}) ⊆ 𝑉)
9290, 91sylib 208 . . . . . . . . . . . . . 14 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ⊆ 𝑉)
9366, 1syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑉𝑊)
9492, 93sstrd 3763 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ⊆ 𝑊)
95 difexg 4943 . . . . . . . . . . . . . . 15 (𝑥 ∈ V → (𝑥 ∖ {𝑋}) ∈ V)
9669, 95ax-mp 5 . . . . . . . . . . . . . 14 (𝑥 ∖ {𝑋}) ∈ V
9796elpw 4304 . . . . . . . . . . . . 13 ((𝑥 ∖ {𝑋}) ∈ 𝒫 𝑊 ↔ (𝑥 ∖ {𝑋}) ⊆ 𝑊)
9894, 97sylibr 224 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ 𝒫 𝑊)
9966, 8syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑆 ∈ Fin)
100 ssfi 8337 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ Fin ∧ 𝑥𝑆) → 𝑥 ∈ Fin)
10199, 68, 100syl2anc 567 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ Fin)
102 diffi 8349 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Fin → (𝑥 ∖ {𝑋}) ∈ Fin)
103101, 102syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ Fin)
104 hashcl 13350 . . . . . . . . . . . . . . 15 ((𝑥 ∖ {𝑋}) ∈ Fin → (♯‘(𝑥 ∖ {𝑋})) ∈ ℕ0)
105 nn0cn 11505 . . . . . . . . . . . . . . 15 ((♯‘(𝑥 ∖ {𝑋})) ∈ ℕ0 → (♯‘(𝑥 ∖ {𝑋})) ∈ ℂ)
106103, 104, 1053syl 18 . . . . . . . . . . . . . 14 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘(𝑥 ∖ {𝑋})) ∈ ℂ)
107 ax-1cn 10197 . . . . . . . . . . . . . 14 1 ∈ ℂ
108 pncan 10490 . . . . . . . . . . . . . 14 (((♯‘(𝑥 ∖ {𝑋})) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (♯‘(𝑥 ∖ {𝑋})))
109106, 107, 108sylancl 568 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (♯‘(𝑥 ∖ {𝑋})))
110 neldifsnd 4460 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑋 ∈ (𝑥 ∖ {𝑋}))
111 hashunsng 13384 . . . . . . . . . . . . . . . . 17 (𝑋𝑆 → (((𝑥 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑥 ∖ {𝑋})) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1)))
11266, 5, 1113syl 18 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((𝑥 ∖ {𝑋}) ∈ Fin ∧ ¬ 𝑋 ∈ (𝑥 ∖ {𝑋})) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1)))
113103, 110, 112mp2and 673 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1))
114 undif1 4186 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∖ {𝑋}) ∪ {𝑋}) = (𝑥 ∪ {𝑋})
115 simpr 471 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑥 ∈ 𝒫 𝑉)
11664, 115eldifd 3735 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝒫 (𝑉 ∪ {𝑋}) ∖ 𝒫 𝑉))
117 elpwunsn 4363 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 (𝑉 ∪ {𝑋}) ∖ 𝒫 𝑉) → 𝑋𝑥)
118116, 117syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑋𝑥)
119118snssd 4476 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → {𝑋} ⊆ 𝑥)
120 ssequn2 3938 . . . . . . . . . . . . . . . . . . 19 ({𝑋} ⊆ 𝑥 ↔ (𝑥 ∪ {𝑋}) = 𝑥)
121119, 120sylib 208 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∪ {𝑋}) = 𝑥)
122114, 121syl5req 2818 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
123122fveq2d 6337 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘𝑥) = (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
124123, 72eqtr3d 2807 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝑀)
125113, 124eqtr3d 2807 . . . . . . . . . . . . . 14 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ((♯‘(𝑥 ∖ {𝑋})) + 1) = 𝑀)
126125oveq1d 6809 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (𝑀 − 1))
127109, 126eqtr3d 2807 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1))
128 fveq2 6333 . . . . . . . . . . . . . 14 (𝑢 = (𝑥 ∖ {𝑋}) → (♯‘𝑢) = (♯‘(𝑥 ∖ {𝑋})))
129128eqeq1d 2773 . . . . . . . . . . . . 13 (𝑢 = (𝑥 ∖ {𝑋}) → ((♯‘𝑢) = (𝑀 − 1) ↔ (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1)))
130129elrab 3516 . . . . . . . . . . . 12 ((𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)} ↔ ((𝑥 ∖ {𝑋}) ∈ 𝒫 𝑊 ∧ (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1)))
13198, 127, 130sylanbrc 566 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
13288, 131sseldd 3754 . . . . . . . . . 10 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ (𝐻 “ {𝐷}))
133 ramub1.h . . . . . . . . . . . 12 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))
134133mptiniseg 5774 . . . . . . . . . . 11 (𝐷𝑅 → (𝐻 “ {𝐷}) = {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷})
13566, 19, 1343syl 18 . . . . . . . . . 10 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐻 “ {𝐷}) = {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷})
136132, 135eleqtrd 2852 . . . . . . . . 9 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷})
137 uneq1 3912 . . . . . . . . . . . . 13 (𝑢 = (𝑥 ∖ {𝑋}) → (𝑢 ∪ {𝑋}) = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
138137fveq2d 6337 . . . . . . . . . . . 12 (𝑢 = (𝑥 ∖ {𝑋}) → (𝐾‘(𝑢 ∪ {𝑋})) = (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
139138eqeq1d 2773 . . . . . . . . . . 11 (𝑢 = (𝑥 ∖ {𝑋}) → ((𝐾‘(𝑢 ∪ {𝑋})) = 𝐷 ↔ (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷))
140139elrab 3516 . . . . . . . . . 10 ((𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷} ↔ ((𝑥 ∖ {𝑋}) ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∧ (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷))
141140simprbi 480 . . . . . . . . 9 ((𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ∣ (𝐾‘(𝑢 ∪ {𝑋})) = 𝐷} → (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷)
142136, 141syl 17 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷)
143122fveq2d 6337 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾𝑥) = (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
144 simpl1r 1280 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝐸 = 𝐷)
145142, 143, 1443eqtr4d 2815 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾𝑥) = 𝐸)
146 ramub1.6 . . . . . . . . 9 (𝜑𝐾:(𝑆𝐶𝑀)⟶𝑅)
147 ffn 6186 . . . . . . . . 9 (𝐾:(𝑆𝐶𝑀)⟶𝑅𝐾 Fn (𝑆𝐶𝑀))
148146, 147syl 17 . . . . . . . 8 (𝜑𝐾 Fn (𝑆𝐶𝑀))
149 fniniseg 6482 . . . . . . . 8 (𝐾 Fn (𝑆𝐶𝑀) → (𝑥 ∈ (𝐾 “ {𝐸}) ↔ (𝑥 ∈ (𝑆𝐶𝑀) ∧ (𝐾𝑥) = 𝐸)))
15066, 148, 1493syl 18 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∈ (𝐾 “ {𝐸}) ↔ (𝑥 ∈ (𝑆𝐶𝑀) ∧ (𝐾𝑥) = 𝐸)))
15178, 145, 150mpbir2and 686 . . . . . 6 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝐾 “ {𝐸}))
15263, 151pm2.61dan 807 . . . . 5 (((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) → 𝑥 ∈ (𝐾 “ {𝐸}))
153152rabssdv 3832 . . . 4 ((𝜑𝐸 = 𝐷) → {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀} ⊆ (𝐾 “ {𝐸}))
15452, 153eqsstrd 3789 . . 3 ((𝜑𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
155 fveq2 6333 . . . . . 6 (𝑧 = (𝑉 ∪ {𝑋}) → (♯‘𝑧) = (♯‘(𝑉 ∪ {𝑋})))
156155breq2d 4799 . . . . 5 (𝑧 = (𝑉 ∪ {𝑋}) → ((𝐹𝐸) ≤ (♯‘𝑧) ↔ (𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋}))))
157 oveq1 6801 . . . . . 6 (𝑧 = (𝑉 ∪ {𝑋}) → (𝑧𝐶𝑀) = ((𝑉 ∪ {𝑋})𝐶𝑀))
158157sseq1d 3782 . . . . 5 (𝑧 = (𝑉 ∪ {𝑋}) → ((𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸}) ↔ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
159156, 158anbi12d 610 . . . 4 (𝑧 = (𝑉 ∪ {𝑋}) → (((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})) ↔ ((𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ∧ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))))
160159rspcev 3461 . . 3 (((𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆 ∧ ((𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ∧ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
16112, 43, 154, 160syl12anc 1474 . 2 ((𝜑𝐸 = 𝐷) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
162 elpw2g 4959 . . . . . 6 (𝑆 ∈ Fin → (𝑉 ∈ 𝒫 𝑆𝑉𝑆))
1638, 162syl 17 . . . . 5 (𝜑 → (𝑉 ∈ 𝒫 𝑆𝑉𝑆))
1644, 163mpbird 247 . . . 4 (𝜑𝑉 ∈ 𝒫 𝑆)
165164adantr 466 . . 3 ((𝜑𝐸𝐷) → 𝑉 ∈ 𝒫 𝑆)
166 ifnefalse 4238 . . . . 5 (𝐸𝐷 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = (𝐹𝐸))
167166adantl 467 . . . 4 ((𝜑𝐸𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = (𝐹𝐸))
16815adantr 466 . . . 4 ((𝜑𝐸𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (♯‘𝑉))
169167, 168eqbrtrrd 4811 . . 3 ((𝜑𝐸𝐷) → (𝐹𝐸) ≤ (♯‘𝑉))
17056adantr 466 . . 3 ((𝜑𝐸𝐷) → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
171 fveq2 6333 . . . . . 6 (𝑧 = 𝑉 → (♯‘𝑧) = (♯‘𝑉))
172171breq2d 4799 . . . . 5 (𝑧 = 𝑉 → ((𝐹𝐸) ≤ (♯‘𝑧) ↔ (𝐹𝐸) ≤ (♯‘𝑉)))
173 oveq1 6801 . . . . . 6 (𝑧 = 𝑉 → (𝑧𝐶𝑀) = (𝑉𝐶𝑀))
174173sseq1d 3782 . . . . 5 (𝑧 = 𝑉 → ((𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸}) ↔ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
175172, 174anbi12d 610 . . . 4 (𝑧 = 𝑉 → (((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})) ↔ ((𝐹𝐸) ≤ (♯‘𝑉) ∧ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))))
176175rspcev 3461 . . 3 ((𝑉 ∈ 𝒫 𝑆 ∧ ((𝐹𝐸) ≤ (♯‘𝑉) ∧ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
177165, 169, 170, 176syl12anc 1474 . 2 ((𝜑𝐸𝐷) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
178161, 177pm2.61dane 3030 1 (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wrex 3062  {crab 3065  Vcvv 3351  cdif 3721  cun 3722  wss 3724  ifcif 4226  𝒫 cpw 4298  {csn 4317   class class class wbr 4787  cmpt 4864  ccnv 5249  cima 5253   Fn wfn 6027  wf 6028  cfv 6032  (class class class)co 6794  cmpt2 6796  Fincfn 8110  cc 10137  cr 10138  1c1 10140   + caddc 10142  cle 10278  cmin 10469  cn 11223  0cn0 11495  chash 13322   Ramsey cram 15911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-cnex 10195  ax-resscn 10196  ax-1cn 10197  ax-icn 10198  ax-addcl 10199  ax-addrcl 10200  ax-mulcl 10201  ax-mulrcl 10202  ax-mulcom 10203  ax-addass 10204  ax-mulass 10205  ax-distr 10206  ax-i2m1 10207  ax-1ne0 10208  ax-1rid 10209  ax-rnegex 10210  ax-rrecex 10211  ax-cnre 10212  ax-pre-lttri 10213  ax-pre-lttrn 10214  ax-pre-ltadd 10215  ax-pre-mulgt0 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-oadd 7718  df-er 7897  df-en 8111  df-dom 8112  df-sdom 8113  df-fin 8114  df-card 8966  df-cda 9193  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282  df-le 10283  df-sub 10471  df-neg 10472  df-nn 11224  df-n0 11496  df-z 11581  df-uz 11890  df-fz 12535  df-hash 13323
This theorem is referenced by:  ramub1lem2  15939
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