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Theorem ramub1lem1 15665
Description: Lemma for ramub1 15667. (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 3597 . . . . . . 7 (𝜑𝑉 ⊆ (𝑆 ∖ {𝑋}))
43difss2d 3723 . . . . . 6 (𝜑𝑉𝑆)
5 ramub1.x . . . . . . 7 (𝜑𝑋𝑆)
65snssd 4314 . . . . . 6 (𝜑 → {𝑋} ⊆ 𝑆)
74, 6unssd 3772 . . . . 5 (𝜑 → (𝑉 ∪ {𝑋}) ⊆ 𝑆)
8 ramub1.4 . . . . . 6 (𝜑𝑆 ∈ Fin)
9 elpw2g 4792 . . . . . 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 481 . . 3 ((𝜑𝐸 = 𝐷) → (𝑉 ∪ {𝑋}) ∈ 𝒫 𝑆)
13 iftrue 4069 . . . . . . 7 (𝐸 = 𝐷 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = ((𝐹𝐷) − 1))
1413adantl 482 . . . . . 6 ((𝜑𝐸 = 𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = ((𝐹𝐷) − 1))
15 ramub1.9 . . . . . . 7 (𝜑 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (#‘𝑉))
1615adantr 481 . . . . . 6 ((𝜑𝐸 = 𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (#‘𝑉))
1714, 16eqbrtrrd 4642 . . . . 5 ((𝜑𝐸 = 𝐷) → ((𝐹𝐷) − 1) ≤ (#‘𝑉))
18 ramub1.f . . . . . . . . 9 (𝜑𝐹:𝑅⟶ℕ)
19 ramub1.d . . . . . . . . 9 (𝜑𝐷𝑅)
2018, 19ffvelrnd 6321 . . . . . . . 8 (𝜑 → (𝐹𝐷) ∈ ℕ)
2120adantr 481 . . . . . . 7 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ∈ ℕ)
2221nnred 10987 . . . . . 6 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ∈ ℝ)
23 1red 10007 . . . . . 6 ((𝜑𝐸 = 𝐷) → 1 ∈ ℝ)
24 ssfi 8132 . . . . . . . . 9 ((𝑆 ∈ Fin ∧ 𝑉𝑆) → 𝑉 ∈ Fin)
258, 4, 24syl2anc 692 . . . . . . . 8 (𝜑𝑉 ∈ Fin)
26 hashcl 13095 . . . . . . . 8 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
27 nn0re 11253 . . . . . . . 8 ((#‘𝑉) ∈ ℕ0 → (#‘𝑉) ∈ ℝ)
2825, 26, 273syl 18 . . . . . . 7 (𝜑 → (#‘𝑉) ∈ ℝ)
2928adantr 481 . . . . . 6 ((𝜑𝐸 = 𝐷) → (#‘𝑉) ∈ ℝ)
3022, 23, 29lesubaddd 10576 . . . . 5 ((𝜑𝐸 = 𝐷) → (((𝐹𝐷) − 1) ≤ (#‘𝑉) ↔ (𝐹𝐷) ≤ ((#‘𝑉) + 1)))
3117, 30mpbid 222 . . . 4 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ≤ ((#‘𝑉) + 1))
32 fveq2 6153 . . . . 5 (𝐸 = 𝐷 → (𝐹𝐸) = (𝐹𝐷))
33 snidg 4182 . . . . . . . 8 (𝑋𝑆𝑋 ∈ {𝑋})
345, 33syl 17 . . . . . . 7 (𝜑𝑋 ∈ {𝑋})
353sseld 3586 . . . . . . . 8 (𝜑 → (𝑋𝑉𝑋 ∈ (𝑆 ∖ {𝑋})))
36 eldifn 3716 . . . . . . . 8 (𝑋 ∈ (𝑆 ∖ {𝑋}) → ¬ 𝑋 ∈ {𝑋})
3735, 36syl6 35 . . . . . . 7 (𝜑 → (𝑋𝑉 → ¬ 𝑋 ∈ {𝑋}))
3834, 37mt2d 131 . . . . . 6 (𝜑 → ¬ 𝑋𝑉)
39 hashunsng 13129 . . . . . . 7 (𝑋𝑆 → ((𝑉 ∈ Fin ∧ ¬ 𝑋𝑉) → (#‘(𝑉 ∪ {𝑋})) = ((#‘𝑉) + 1)))
405, 39syl 17 . . . . . 6 (𝜑 → ((𝑉 ∈ Fin ∧ ¬ 𝑋𝑉) → (#‘(𝑉 ∪ {𝑋})) = ((#‘𝑉) + 1)))
4125, 38, 40mp2and 714 . . . . 5 (𝜑 → (#‘(𝑉 ∪ {𝑋})) = ((#‘𝑉) + 1))
4232, 41breqan12rd 4635 . . . 4 ((𝜑𝐸 = 𝐷) → ((𝐹𝐸) ≤ (#‘(𝑉 ∪ {𝑋})) ↔ (𝐹𝐷) ≤ ((#‘𝑉) + 1)))
4331, 42mpbird 247 . . 3 ((𝜑𝐸 = 𝐷) → (𝐹𝐸) ≤ (#‘(𝑉 ∪ {𝑋})))
44 snfi 7990 . . . . . . 7 {𝑋} ∈ Fin
45 unfi 8179 . . . . . . 7 ((𝑉 ∈ Fin ∧ {𝑋} ∈ Fin) → (𝑉 ∪ {𝑋}) ∈ Fin)
4625, 44, 45sylancl 693 . . . . . 6 (𝜑 → (𝑉 ∪ {𝑋}) ∈ Fin)
47 ramub1.m . . . . . . 7 (𝜑𝑀 ∈ ℕ)
4847nnnn0d 11303 . . . . . 6 (𝜑𝑀 ∈ ℕ0)
49 ramub1.3 . . . . . . 7 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
5049hashbcval 15641 . . . . . 6 (((𝑉 ∪ {𝑋}) ∈ Fin ∧ 𝑀 ∈ ℕ0) → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (#‘𝑥) = 𝑀})
5146, 48, 50syl2anc 692 . . . . 5 (𝜑 → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (#‘𝑥) = 𝑀})
5251adantr 481 . . . 4 ((𝜑𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (#‘𝑥) = 𝑀})
53 simpl1l 1110 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝜑)
5449hashbcval 15641 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑀 ∈ ℕ0) → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 𝑀})
5525, 48, 54syl2anc 692 . . . . . . . . 9 (𝜑 → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 𝑀})
56 ramub1.s . . . . . . . . 9 (𝜑 → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
5755, 56eqsstr3d 3624 . . . . . . . 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 477 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 𝑉)
60 simpl3 1064 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → (#‘𝑥) = 𝑀)
61 rabid 3109 . . . . . . . 8 (𝑥 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 𝑀} ↔ (𝑥 ∈ 𝒫 𝑉 ∧ (#‘𝑥) = 𝑀))
6259, 60, 61sylanbrc 697 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 𝑀})
6358, 62sseldd 3588 . . . . . 6 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝐾 “ {𝐸}))
64 simpl2 1063 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}))
6564elpwid 4146 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ (𝑉 ∪ {𝑋}))
66 simpl1l 1110 . . . . . . . . . . . 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 3597 . . . . . . . . . 10 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥𝑆)
69 vex 3192 . . . . . . . . . . 11 𝑥 ∈ V
7069elpw 4141 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝑆𝑥𝑆)
7168, 70sylibr 224 . . . . . . . . 9 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ 𝒫 𝑆)
72 simpl3 1064 . . . . . . . . 9 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (#‘𝑥) = 𝑀)
73 rabid 3109 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (#‘𝑥) = 𝑀} ↔ (𝑥 ∈ 𝒫 𝑆 ∧ (#‘𝑥) = 𝑀))
7471, 72, 73sylanbrc 697 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (#‘𝑥) = 𝑀})
7549hashbcval 15641 . . . . . . . . . 10 ((𝑆 ∈ Fin ∧ 𝑀 ∈ ℕ0) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (#‘𝑥) = 𝑀})
768, 48, 75syl2anc 692 . . . . . . . . 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 2701 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝑆𝐶𝑀))
792difss2d 3723 . . . . . . . . . . . . . . 15 (𝜑𝑊𝑆)
80 ssfi 8132 . . . . . . . . . . . . . . 15 ((𝑆 ∈ Fin ∧ 𝑊𝑆) → 𝑊 ∈ Fin)
818, 79, 80syl2anc 692 . . . . . . . . . . . . . 14 (𝜑𝑊 ∈ Fin)
82 nnm1nn0 11286 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
8347, 82syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑀 − 1) ∈ ℕ0)
8449hashbcval 15641 . . . . . . . . . . . . . 14 ((𝑊 ∈ Fin ∧ (𝑀 − 1) ∈ ℕ0) → (𝑊𝐶(𝑀 − 1)) = {𝑢 ∈ 𝒫 𝑊 ∣ (#‘𝑢) = (𝑀 − 1)})
8581, 83, 84syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐶(𝑀 − 1)) = {𝑢 ∈ 𝒫 𝑊 ∣ (#‘𝑢) = (𝑀 − 1)})
86 ramub1.8 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝐷}))
8785, 86eqsstr3d 3624 . . . . . . . . . . . 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 3740 . . . . . . . . . . . . . . . 16 (𝑉 ∪ {𝑋}) = ({𝑋} ∪ 𝑉)
9065, 89syl6sseq 3635 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ ({𝑋} ∪ 𝑉))
91 ssundif 4029 . . . . . . . . . . . . . . 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 3597 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ⊆ 𝑊)
95 difexg 4773 . . . . . . . . . . . . . . 15 (𝑥 ∈ V → (𝑥 ∖ {𝑋}) ∈ V)
9669, 95ax-mp 5 . . . . . . . . . . . . . 14 (𝑥 ∖ {𝑋}) ∈ V
9796elpw 4141 . . . . . . . . . . . . 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 8132 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ Fin ∧ 𝑥𝑆) → 𝑥 ∈ Fin)
10199, 68, 100syl2anc 692 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ Fin)
102 diffi 8144 . . . . . . . . . . . . . . . 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 13095 . . . . . . . . . . . . . . 15 ((𝑥 ∖ {𝑋}) ∈ Fin → (#‘(𝑥 ∖ {𝑋})) ∈ ℕ0)
105 nn0cn 11254 . . . . . . . . . . . . . . 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 9946 . . . . . . . . . . . . . 14 1 ∈ ℂ
108 pncan 10239 . . . . . . . . . . . . . 14 (((#‘(𝑥 ∖ {𝑋})) ∈ ℂ ∧ 1 ∈ ℂ) → (((#‘(𝑥 ∖ {𝑋})) + 1) − 1) = (#‘(𝑥 ∖ {𝑋})))
109106, 107, 108sylancl 693 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((#‘(𝑥 ∖ {𝑋})) + 1) − 1) = (#‘(𝑥 ∖ {𝑋})))
110 neldifsnd 4296 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑋 ∈ (𝑥 ∖ {𝑋}))
111 hashunsng 13129 . . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (#‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((#‘(𝑥 ∖ {𝑋})) + 1))
114 undif1 4020 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∖ {𝑋}) ∪ {𝑋}) = (𝑥 ∪ {𝑋})
115 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑥 ∈ 𝒫 𝑉)
11664, 115eldifd 3570 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝒫 (𝑉 ∪ {𝑋}) ∖ 𝒫 𝑉))
117 elpwunsn 4200 . . . . . . . . . . . . . . . . . . . . 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 4314 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → {𝑋} ⊆ 𝑥)
120 ssequn2 3769 . . . . . . . . . . . . . . . . . . 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 2668 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
123122fveq2d 6157 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (#‘𝑥) = (#‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
124123, 72eqtr3d 2657 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (#‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝑀)
125113, 124eqtr3d 2657 . . . . . . . . . . . . . 14 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ((#‘(𝑥 ∖ {𝑋})) + 1) = 𝑀)
126125oveq1d 6625 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((#‘(𝑥 ∖ {𝑋})) + 1) − 1) = (𝑀 − 1))
127109, 126eqtr3d 2657 . . . . . . . . . . . 12 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (#‘(𝑥 ∖ {𝑋})) = (𝑀 − 1))
128 fveq2 6153 . . . . . . . . . . . . . 14 (𝑢 = (𝑥 ∖ {𝑋}) → (#‘𝑢) = (#‘(𝑥 ∖ {𝑋})))
129128eqeq1d 2623 . . . . . . . . . . . . 13 (𝑢 = (𝑥 ∖ {𝑋}) → ((#‘𝑢) = (𝑀 − 1) ↔ (#‘(𝑥 ∖ {𝑋})) = (𝑀 − 1)))
130129elrab 3350 . . . . . . . . . . . 12 ((𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ 𝒫 𝑊 ∣ (#‘𝑢) = (𝑀 − 1)} ↔ ((𝑥 ∖ {𝑋}) ∈ 𝒫 𝑊 ∧ (#‘(𝑥 ∖ {𝑋})) = (𝑀 − 1)))
13198, 127, 130sylanbrc 697 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ 𝒫 𝑊 ∣ (#‘𝑢) = (𝑀 − 1)})
13288, 131sseldd 3588 . . . . . . . . . 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 5593 . . . . . . . . . . 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 2700 . . . . . . . . 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 3743 . . . . . . . . . . . . 13 (𝑢 = (𝑥 ∖ {𝑋}) → (𝑢 ∪ {𝑋}) = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
138137fveq2d 6157 . . . . . . . . . . . 12 (𝑢 = (𝑥 ∖ {𝑋}) → (𝐾‘(𝑢 ∪ {𝑋})) = (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
139138eqeq1d 2623 . . . . . . . . . . 11 (𝑢 = (𝑥 ∖ {𝑋}) → ((𝐾‘(𝑢 ∪ {𝑋})) = 𝐷 ↔ (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = 𝐷))
140139elrab 3350 . . . . . . . . . 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 6157 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾𝑥) = (𝐾‘((𝑥 ∖ {𝑋}) ∪ {𝑋})))
144 simpl1r 1111 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝐸 = 𝐷)
145142, 143, 1443eqtr4d 2665 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝐾𝑥) = 𝐸)
146 ramub1.6 . . . . . . . . 9 (𝜑𝐾:(𝑆𝐶𝑀)⟶𝑅)
147 ffn 6007 . . . . . . . . 9 (𝐾:(𝑆𝐶𝑀)⟶𝑅𝐾 Fn (𝑆𝐶𝑀))
148146, 147syl 17 . . . . . . . 8 (𝜑𝐾 Fn (𝑆𝐶𝑀))
149 fniniseg 6299 . . . . . . . 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 956 . . . . . 6 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝐾 “ {𝐸}))
15263, 151pm2.61dan 831 . . . . 5 (((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (#‘𝑥) = 𝑀) → 𝑥 ∈ (𝐾 “ {𝐸}))
153152rabssdv 3666 . . . 4 ((𝜑𝐸 = 𝐷) → {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (#‘𝑥) = 𝑀} ⊆ (𝐾 “ {𝐸}))
15452, 153eqsstrd 3623 . . 3 ((𝜑𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
155 fveq2 6153 . . . . . 6 (𝑧 = (𝑉 ∪ {𝑋}) → (#‘𝑧) = (#‘(𝑉 ∪ {𝑋})))
156155breq2d 4630 . . . . 5 (𝑧 = (𝑉 ∪ {𝑋}) → ((𝐹𝐸) ≤ (#‘𝑧) ↔ (𝐹𝐸) ≤ (#‘(𝑉 ∪ {𝑋}))))
157 oveq1 6617 . . . . . 6 (𝑧 = (𝑉 ∪ {𝑋}) → (𝑧𝐶𝑀) = ((𝑉 ∪ {𝑋})𝐶𝑀))
158157sseq1d 3616 . . . . 5 (𝑧 = (𝑉 ∪ {𝑋}) → ((𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸}) ↔ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
159156, 158anbi12d 746 . . . 4 (𝑧 = (𝑉 ∪ {𝑋}) → (((𝐹𝐸) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})) ↔ ((𝐹𝐸) ≤ (#‘(𝑉 ∪ {𝑋})) ∧ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))))
160159rspcev 3298 . . 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 1321 . 2 ((𝜑𝐸 = 𝐷) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
162 elpw2g 4792 . . . . . 6 (𝑆 ∈ Fin → (𝑉 ∈ 𝒫 𝑆𝑉𝑆))
1638, 162syl 17 . . . . 5 (𝜑 → (𝑉 ∈ 𝒫 𝑆𝑉𝑆))
1644, 163mpbird 247 . . . 4 (𝜑𝑉 ∈ 𝒫 𝑆)
165164adantr 481 . . 3 ((𝜑𝐸𝐷) → 𝑉 ∈ 𝒫 𝑆)
166 ifnefalse 4075 . . . . 5 (𝐸𝐷 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = (𝐹𝐸))
167166adantl 482 . . . 4 ((𝜑𝐸𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) = (𝐹𝐸))
16815adantr 481 . . . 4 ((𝜑𝐸𝐷) → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (#‘𝑉))
169167, 168eqbrtrrd 4642 . . 3 ((𝜑𝐸𝐷) → (𝐹𝐸) ≤ (#‘𝑉))
17056adantr 481 . . 3 ((𝜑𝐸𝐷) → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
171 fveq2 6153 . . . . . 6 (𝑧 = 𝑉 → (#‘𝑧) = (#‘𝑉))
172171breq2d 4630 . . . . 5 (𝑧 = 𝑉 → ((𝐹𝐸) ≤ (#‘𝑧) ↔ (𝐹𝐸) ≤ (#‘𝑉)))
173 oveq1 6617 . . . . . 6 (𝑧 = 𝑉 → (𝑧𝐶𝑀) = (𝑉𝐶𝑀))
174173sseq1d 3616 . . . . 5 (𝑧 = 𝑉 → ((𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸}) ↔ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
175172, 174anbi12d 746 . . . 4 (𝑧 = 𝑉 → (((𝐹𝐸) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})) ↔ ((𝐹𝐸) ≤ (#‘𝑉) ∧ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))))
176175rspcev 3298 . . 3 ((𝑉 ∈ 𝒫 𝑆 ∧ ((𝐹𝐸) ≤ (#‘𝑉) ∧ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
177165, 169, 170, 176syl12anc 1321 . 2 ((𝜑𝐸𝐷) → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
178161, 177pm2.61dane 2877 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908  {crab 2911  Vcvv 3189  cdif 3556  cun 3557  wss 3559  ifcif 4063  𝒫 cpw 4135  {csn 4153   class class class wbr 4618  cmpt 4678  ccnv 5078  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  Fincfn 7907  cc 9886  cr 9887  1c1 9889   + caddc 9891  cle 10027  cmin 10218  cn 10972  0cn0 11244  #chash 13065   Ramsey cram 15638
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-hash 13066
This theorem is referenced by:  ramub1lem2  15666
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