MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ramub1lem1 Structured version   Visualization version   GIF version

Theorem ramub1lem1 15937
Description: Lemma for ramub1 15939. (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 3762 . . . . . . 7 (𝜑𝑉 ⊆ (𝑆 ∖ {𝑋}))
43difss2d 3891 . . . . . 6 (𝜑𝑉𝑆)
5 ramub1.x . . . . . . 7 (𝜑𝑋𝑆)
65snssd 4475 . . . . . 6 (𝜑 → {𝑋} ⊆ 𝑆)
74, 6unssd 3940 . . . . 5 (𝜑 → (𝑉 ∪ {𝑋}) ⊆ 𝑆)
8 ramub1.4 . . . . . 6 (𝜑𝑆 ∈ Fin)
9 elpw2g 4958 . . . . . 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 4231 . . . . . . 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 4810 . . . . 5 ((𝜑𝐸 = 𝐷) → ((𝐹𝐷) − 1) ≤ (♯‘𝑉))
18 ramub1.f . . . . . . . . 9 (𝜑𝐹:𝑅⟶ℕ)
19 ramub1.d . . . . . . . . 9 (𝜑𝐷𝑅)
2018, 19ffvelrnd 6503 . . . . . . . 8 (𝜑 → (𝐹𝐷) ∈ ℕ)
2120adantr 466 . . . . . . 7 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ∈ ℕ)
2221nnred 11237 . . . . . 6 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ∈ ℝ)
23 1red 10257 . . . . . 6 ((𝜑𝐸 = 𝐷) → 1 ∈ ℝ)
24 ssfi 8336 . . . . . . . . 9 ((𝑆 ∈ Fin ∧ 𝑉𝑆) → 𝑉 ∈ Fin)
258, 4, 24syl2anc 573 . . . . . . . 8 (𝜑𝑉 ∈ Fin)
26 hashcl 13349 . . . . . . . 8 (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
27 nn0re 11503 . . . . . . . 8 ((♯‘𝑉) ∈ ℕ0 → (♯‘𝑉) ∈ ℝ)
2825, 26, 273syl 18 . . . . . . 7 (𝜑 → (♯‘𝑉) ∈ ℝ)
2928adantr 466 . . . . . 6 ((𝜑𝐸 = 𝐷) → (♯‘𝑉) ∈ ℝ)
3022, 23, 29lesubaddd 10826 . . . . 5 ((𝜑𝐸 = 𝐷) → (((𝐹𝐷) − 1) ≤ (♯‘𝑉) ↔ (𝐹𝐷) ≤ ((♯‘𝑉) + 1)))
3117, 30mpbid 222 . . . 4 ((𝜑𝐸 = 𝐷) → (𝐹𝐷) ≤ ((♯‘𝑉) + 1))
32 fveq2 6332 . . . . 5 (𝐸 = 𝐷 → (𝐹𝐸) = (𝐹𝐷))
33 snidg 4345 . . . . . . . 8 (𝑋𝑆𝑋 ∈ {𝑋})
345, 33syl 17 . . . . . . 7 (𝜑𝑋 ∈ {𝑋})
353sseld 3751 . . . . . . . 8 (𝜑 → (𝑋𝑉𝑋 ∈ (𝑆 ∖ {𝑋})))
36 eldifn 3884 . . . . . . . 8 (𝑋 ∈ (𝑆 ∖ {𝑋}) → ¬ 𝑋 ∈ {𝑋})
3735, 36syl6 35 . . . . . . 7 (𝜑 → (𝑋𝑉 → ¬ 𝑋 ∈ {𝑋}))
3834, 37mt2d 133 . . . . . 6 (𝜑 → ¬ 𝑋𝑉)
39 hashunsng 13383 . . . . . . 7 (𝑋𝑆 → ((𝑉 ∈ Fin ∧ ¬ 𝑋𝑉) → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1)))
405, 39syl 17 . . . . . 6 (𝜑 → ((𝑉 ∈ Fin ∧ ¬ 𝑋𝑉) → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1)))
4125, 38, 40mp2and 679 . . . . 5 (𝜑 → (♯‘(𝑉 ∪ {𝑋})) = ((♯‘𝑉) + 1))
4232, 41breqan12rd 4803 . . . 4 ((𝜑𝐸 = 𝐷) → ((𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ↔ (𝐹𝐷) ≤ ((♯‘𝑉) + 1)))
4331, 42mpbird 247 . . 3 ((𝜑𝐸 = 𝐷) → (𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})))
44 snfi 8194 . . . . . . 7 {𝑋} ∈ Fin
45 unfi 8383 . . . . . . 7 ((𝑉 ∈ Fin ∧ {𝑋} ∈ Fin) → (𝑉 ∪ {𝑋}) ∈ Fin)
4625, 44, 45sylancl 574 . . . . . 6 (𝜑 → (𝑉 ∪ {𝑋}) ∈ Fin)
47 ramub1.m . . . . . . 7 (𝜑𝑀 ∈ ℕ)
4847nnnn0d 11553 . . . . . 6 (𝜑𝑀 ∈ ℕ0)
49 ramub1.3 . . . . . . 7 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
5049hashbcval 15913 . . . . . 6 (((𝑉 ∪ {𝑋}) ∈ Fin ∧ 𝑀 ∈ ℕ0) → ((𝑉 ∪ {𝑋})𝐶𝑀) = {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀})
5146, 48, 50syl2anc 573 . . . . 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 15913 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑀 ∈ ℕ0) → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
5525, 48, 54syl2anc 573 . . . . . . . . 9 (𝜑 → (𝑉𝐶𝑀) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
56 ramub1.s . . . . . . . . 9 (𝜑 → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
5755, 56eqsstr3d 3789 . . . . . . . 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 572 . . . . . . 7 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 𝑀})
6358, 62sseldd 3753 . . . . . 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 4309 . . . . . . . . . . 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 3762 . . . . . . . . . 10 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥𝑆)
69 vex 3354 . . . . . . . . . . 11 𝑥 ∈ V
7069elpw 4303 . . . . . . . . . 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 572 . . . . . . . 8 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
7549hashbcval 15913 . . . . . . . . . 10 ((𝑆 ∈ Fin ∧ 𝑀 ∈ ℕ0) → (𝑆𝐶𝑀) = {𝑥 ∈ 𝒫 𝑆 ∣ (♯‘𝑥) = 𝑀})
768, 48, 75syl2anc 573 . . . . . . . . 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 3891 . . . . . . . . . . . . . . 15 (𝜑𝑊𝑆)
80 ssfi 8336 . . . . . . . . . . . . . . 15 ((𝑆 ∈ Fin ∧ 𝑊𝑆) → 𝑊 ∈ Fin)
818, 79, 80syl2anc 573 . . . . . . . . . . . . . 14 (𝜑𝑊 ∈ Fin)
82 nnm1nn0 11536 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
8347, 82syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑀 − 1) ∈ ℕ0)
8449hashbcval 15913 . . . . . . . . . . . . . 14 ((𝑊 ∈ Fin ∧ (𝑀 − 1) ∈ ℕ0) → (𝑊𝐶(𝑀 − 1)) = {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
8581, 83, 84syl2anc 573 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐶(𝑀 − 1)) = {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
86 ramub1.8 . . . . . . . . . . . . 13 (𝜑 → (𝑊𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝐷}))
8785, 86eqsstr3d 3789 . . . . . . . . . . . 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 3908 . . . . . . . . . . . . . . . 16 (𝑉 ∪ {𝑋}) = ({𝑋} ∪ 𝑉)
9065, 89syl6sseq 3800 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ⊆ ({𝑋} ∪ 𝑉))
91 ssundif 4194 . . . . . . . . . . . . . . 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 3762 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ⊆ 𝑊)
95 difexg 4942 . . . . . . . . . . . . . . 15 (𝑥 ∈ V → (𝑥 ∖ {𝑋}) ∈ V)
9669, 95ax-mp 5 . . . . . . . . . . . . . 14 (𝑥 ∖ {𝑋}) ∈ V
9796elpw 4303 . . . . . . . . . . . . 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 8336 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ Fin ∧ 𝑥𝑆) → 𝑥 ∈ Fin)
10199, 68, 100syl2anc 573 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ Fin)
102 diffi 8348 . . . . . . . . . . . . . . . 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 13349 . . . . . . . . . . . . . . 15 ((𝑥 ∖ {𝑋}) ∈ Fin → (♯‘(𝑥 ∖ {𝑋})) ∈ ℕ0)
105 nn0cn 11504 . . . . . . . . . . . . . . 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 10196 . . . . . . . . . . . . . 14 1 ∈ ℂ
108 pncan 10489 . . . . . . . . . . . . . 14 (((♯‘(𝑥 ∖ {𝑋})) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (♯‘(𝑥 ∖ {𝑋})))
109106, 107, 108sylancl 574 . . . . . . . . . . . . 13 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (((♯‘(𝑥 ∖ {𝑋})) + 1) − 1) = (♯‘(𝑥 ∖ {𝑋})))
110 neldifsnd 4459 . . . . . . . . . . . . . . . 16 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → ¬ 𝑋 ∈ (𝑥 ∖ {𝑋}))
111 hashunsng 13383 . . . . . . . . . . . . . . . . 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 679 . . . . . . . . . . . . . . 15 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (♯‘((𝑥 ∖ {𝑋}) ∪ {𝑋})) = ((♯‘(𝑥 ∖ {𝑋})) + 1))
114 undif1 4185 . . . . . . . . . . . . . . . . . 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 3734 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝒫 (𝑉 ∪ {𝑋}) ∖ 𝒫 𝑉))
117 elpwunsn 4362 . . . . . . . . . . . . . . . . . . . . 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 4475 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → {𝑋} ⊆ 𝑥)
120 ssequn2 3937 . . . . . . . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . . . 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 6808 . . . . . . . . . . . . 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 6332 . . . . . . . . . . . . . 14 (𝑢 = (𝑥 ∖ {𝑋}) → (♯‘𝑢) = (♯‘(𝑥 ∖ {𝑋})))
129128eqeq1d 2773 . . . . . . . . . . . . 13 (𝑢 = (𝑥 ∖ {𝑋}) → ((♯‘𝑢) = (𝑀 − 1) ↔ (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1)))
130129elrab 3515 . . . . . . . . . . . 12 ((𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)} ↔ ((𝑥 ∖ {𝑋}) ∈ 𝒫 𝑊 ∧ (♯‘(𝑥 ∖ {𝑋})) = (𝑀 − 1)))
13198, 127, 130sylanbrc 572 . . . . . . . . . . 11 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → (𝑥 ∖ {𝑋}) ∈ {𝑢 ∈ 𝒫 𝑊 ∣ (♯‘𝑢) = (𝑀 − 1)})
13288, 131sseldd 3753 . . . . . . . . . 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 5773 . . . . . . . . . . 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 3911 . . . . . . . . . . . . 13 (𝑢 = (𝑥 ∖ {𝑋}) → (𝑢 ∪ {𝑋}) = ((𝑥 ∖ {𝑋}) ∪ {𝑋}))
138137fveq2d 6336 . . . . . . . . . . . 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 3515 . . . . . . . . . 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 484 . . . . . . . . 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 6336 . . . . . . . 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 6185 . . . . . . . . 9 (𝐾:(𝑆𝐶𝑀)⟶𝑅𝐾 Fn (𝑆𝐶𝑀))
148146, 147syl 17 . . . . . . . 8 (𝜑𝐾 Fn (𝑆𝐶𝑀))
149 fniniseg 6481 . . . . . . . 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 692 . . . . . 6 ((((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) ∧ ¬ 𝑥 ∈ 𝒫 𝑉) → 𝑥 ∈ (𝐾 “ {𝐸}))
15263, 151pm2.61dan 814 . . . . 5 (((𝜑𝐸 = 𝐷) ∧ 𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∧ (♯‘𝑥) = 𝑀) → 𝑥 ∈ (𝐾 “ {𝐸}))
153152rabssdv 3831 . . . 4 ((𝜑𝐸 = 𝐷) → {𝑥 ∈ 𝒫 (𝑉 ∪ {𝑋}) ∣ (♯‘𝑥) = 𝑀} ⊆ (𝐾 “ {𝐸}))
15452, 153eqsstrd 3788 . . 3 ((𝜑𝐸 = 𝐷) → ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
155 fveq2 6332 . . . . . 6 (𝑧 = (𝑉 ∪ {𝑋}) → (♯‘𝑧) = (♯‘(𝑉 ∪ {𝑋})))
156155breq2d 4798 . . . . 5 (𝑧 = (𝑉 ∪ {𝑋}) → ((𝐹𝐸) ≤ (♯‘𝑧) ↔ (𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋}))))
157 oveq1 6800 . . . . . 6 (𝑧 = (𝑉 ∪ {𝑋}) → (𝑧𝐶𝑀) = ((𝑉 ∪ {𝑋})𝐶𝑀))
158157sseq1d 3781 . . . . 5 (𝑧 = (𝑉 ∪ {𝑋}) → ((𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸}) ↔ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
159156, 158anbi12d 616 . . . 4 (𝑧 = (𝑉 ∪ {𝑋}) → (((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})) ↔ ((𝐹𝐸) ≤ (♯‘(𝑉 ∪ {𝑋})) ∧ ((𝑉 ∪ {𝑋})𝐶𝑀) ⊆ (𝐾 “ {𝐸}))))
160159rspcev 3460 . . 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 4958 . . . . . 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 4237 . . . . 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 4810 . . 3 ((𝜑𝐸𝐷) → (𝐹𝐸) ≤ (♯‘𝑉))
17056adantr 466 . . 3 ((𝜑𝐸𝐷) → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))
171 fveq2 6332 . . . . . 6 (𝑧 = 𝑉 → (♯‘𝑧) = (♯‘𝑉))
172171breq2d 4798 . . . . 5 (𝑧 = 𝑉 → ((𝐹𝐸) ≤ (♯‘𝑧) ↔ (𝐹𝐸) ≤ (♯‘𝑉)))
173 oveq1 6800 . . . . . 6 (𝑧 = 𝑉 → (𝑧𝐶𝑀) = (𝑉𝐶𝑀))
174173sseq1d 3781 . . . . 5 (𝑧 = 𝑉 → ((𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸}) ↔ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
175172, 174anbi12d 616 . . . 4 (𝑧 = 𝑉 → (((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})) ↔ ((𝐹𝐸) ≤ (♯‘𝑉) ∧ (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))))
176175rspcev 3460 . . 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 3720  cun 3721  wss 3723  ifcif 4225  𝒫 cpw 4297  {csn 4316   class class class wbr 4786  cmpt 4863  ccnv 5248  cima 5252   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  cmpt2 6795  Fincfn 8109  cc 10136  cr 10137  1c1 10139   + caddc 10141  cle 10277  cmin 10468  cn 11222  0cn0 11494  chash 13321   Ramsey cram 15910
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 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-hash 13322
This theorem is referenced by:  ramub1lem2  15938
  Copyright terms: Public domain W3C validator