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Theorem vdwlem13 16399
 Description: Lemma for vdw 16400. Main induction on 𝐾; 𝐾 = 0, 𝐾 = 1 base cases. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdw.r (𝜑𝑅 ∈ Fin)
vdw.k (𝜑𝐾 ∈ ℕ0)
Assertion
Ref Expression
vdwlem13 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓)
Distinct variable groups:   𝜑,𝑛,𝑓   𝑓,𝐾,𝑛   𝑅,𝑓,𝑛   𝜑,𝑓

Proof of Theorem vdwlem13
Dummy variables 𝑎 𝑐 𝑑 𝑔 𝑘 𝑚 𝑥 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn1uz2 12379 . . 3 (𝐾 ∈ ℕ ↔ (𝐾 = 1 ∨ 𝐾 ∈ (ℤ‘2)))
2 vdw.r . . . . . . . . . 10 (𝜑𝑅 ∈ Fin)
3 ovex 7190 . . . . . . . . . 10 (1...1) ∈ V
4 elmapg 8436 . . . . . . . . . 10 ((𝑅 ∈ Fin ∧ (1...1) ∈ V) → (𝑓 ∈ (𝑅m (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
52, 3, 4sylancl 589 . . . . . . . . 9 (𝜑 → (𝑓 ∈ (𝑅m (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
65biimpa 480 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑅m (1...1))) → 𝑓:(1...1)⟶𝑅)
7 1nn 11699 . . . . . . . . . 10 1 ∈ ℕ
8 vdwap1 16383 . . . . . . . . . 10 ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘1)1) = {1})
97, 7, 8mp2an 691 . . . . . . . . 9 (1(AP‘1)1) = {1}
10 1z 12065 . . . . . . . . . . . 12 1 ∈ ℤ
11 elfz3 12980 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ (1...1))
1210, 11mp1i 13 . . . . . . . . . . 11 ((𝜑𝑓:(1...1)⟶𝑅) → 1 ∈ (1...1))
13 eqidd 2760 . . . . . . . . . . 11 ((𝜑𝑓:(1...1)⟶𝑅) → (𝑓‘1) = (𝑓‘1))
14 ffn 6504 . . . . . . . . . . . . 13 (𝑓:(1...1)⟶𝑅𝑓 Fn (1...1))
1514adantl 485 . . . . . . . . . . . 12 ((𝜑𝑓:(1...1)⟶𝑅) → 𝑓 Fn (1...1))
16 fniniseg 6827 . . . . . . . . . . . 12 (𝑓 Fn (1...1) → (1 ∈ (𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧ (𝑓‘1) = (𝑓‘1))))
1715, 16syl 17 . . . . . . . . . . 11 ((𝜑𝑓:(1...1)⟶𝑅) → (1 ∈ (𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧ (𝑓‘1) = (𝑓‘1))))
1812, 13, 17mpbir2and 712 . . . . . . . . . 10 ((𝜑𝑓:(1...1)⟶𝑅) → 1 ∈ (𝑓 “ {(𝑓‘1)}))
1918snssd 4703 . . . . . . . . 9 ((𝜑𝑓:(1...1)⟶𝑅) → {1} ⊆ (𝑓 “ {(𝑓‘1)}))
209, 19eqsstrid 3943 . . . . . . . 8 ((𝜑𝑓:(1...1)⟶𝑅) → (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
216, 20syldan 594 . . . . . . 7 ((𝜑𝑓 ∈ (𝑅m (1...1))) → (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
2221ralrimiva 3114 . . . . . 6 (𝜑 → ∀𝑓 ∈ (𝑅m (1...1))(1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
23 fveq2 6664 . . . . . . . . 9 (𝐾 = 1 → (AP‘𝐾) = (AP‘1))
2423oveqd 7174 . . . . . . . 8 (𝐾 = 1 → (1(AP‘𝐾)1) = (1(AP‘1)1))
2524sseq1d 3926 . . . . . . 7 (𝐾 = 1 → ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)})))
2625ralbidv 3127 . . . . . 6 (𝐾 = 1 → (∀𝑓 ∈ (𝑅m (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ ∀𝑓 ∈ (𝑅m (1...1))(1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)})))
2722, 26syl5ibrcom 250 . . . . 5 (𝜑 → (𝐾 = 1 → ∀𝑓 ∈ (𝑅m (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)})))
28 oveq1 7164 . . . . . . . . . . . 12 (𝑎 = 1 → (𝑎(AP‘𝐾)𝑑) = (1(AP‘𝐾)𝑑))
2928sseq1d 3926 . . . . . . . . . . 11 (𝑎 = 1 → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)})))
30 oveq2 7165 . . . . . . . . . . . 12 (𝑑 = 1 → (1(AP‘𝐾)𝑑) = (1(AP‘𝐾)1))
3130sseq1d 3926 . . . . . . . . . . 11 (𝑑 = 1 → ((1(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)})))
3229, 31rspc2ev 3556 . . . . . . . . . 10 ((1 ∈ ℕ ∧ 1 ∈ ℕ ∧ (1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}))
337, 7, 32mp3an12 1449 . . . . . . . . 9 ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}))
34 fvex 6677 . . . . . . . . . 10 (𝑓‘1) ∈ V
35 sneq 4536 . . . . . . . . . . . . 13 (𝑐 = (𝑓‘1) → {𝑐} = {(𝑓‘1)})
3635imaeq2d 5907 . . . . . . . . . . . 12 (𝑐 = (𝑓‘1) → (𝑓 “ {𝑐}) = (𝑓 “ {(𝑓‘1)}))
3736sseq2d 3927 . . . . . . . . . . 11 (𝑐 = (𝑓‘1) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)})))
38372rexbidv 3225 . . . . . . . . . 10 (𝑐 = (𝑓‘1) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)})))
3934, 38spcev 3528 . . . . . . . . 9 (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐}))
4033, 39syl 17 . . . . . . . 8 ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐}))
41 vdw.k . . . . . . . . . 10 (𝜑𝐾 ∈ ℕ0)
4241adantr 484 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝑅m (1...1))) → 𝐾 ∈ ℕ0)
433, 42, 6vdwmc 16384 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑅m (1...1))) → (𝐾 MonoAP 𝑓 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐})))
4440, 43syl5ibr 249 . . . . . . 7 ((𝜑𝑓 ∈ (𝑅m (1...1))) → ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → 𝐾 MonoAP 𝑓))
4544ralimdva 3109 . . . . . 6 (𝜑 → (∀𝑓 ∈ (𝑅m (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∀𝑓 ∈ (𝑅m (1...1))𝐾 MonoAP 𝑓))
46 oveq2 7165 . . . . . . . . . 10 (𝑛 = 1 → (1...𝑛) = (1...1))
4746oveq2d 7173 . . . . . . . . 9 (𝑛 = 1 → (𝑅m (1...𝑛)) = (𝑅m (1...1)))
4847raleqdv 3330 . . . . . . . 8 (𝑛 = 1 → (∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅m (1...1))𝐾 MonoAP 𝑓))
4948rspcev 3544 . . . . . . 7 ((1 ∈ ℕ ∧ ∀𝑓 ∈ (𝑅m (1...1))𝐾 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓)
507, 49mpan 689 . . . . . 6 (∀𝑓 ∈ (𝑅m (1...1))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓)
5145, 50syl6 35 . . . . 5 (𝜑 → (∀𝑓 ∈ (𝑅m (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
5227, 51syld 47 . . . 4 (𝜑 → (𝐾 = 1 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
53 breq1 5040 . . . . . . . 8 (𝑥 = 2 → (𝑥 MonoAP 𝑓 ↔ 2 MonoAP 𝑓))
5453rexralbidv 3226 . . . . . . 7 (𝑥 = 2 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓))
5554ralbidv 3127 . . . . . 6 (𝑥 = 2 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓))
56 breq1 5040 . . . . . . . 8 (𝑥 = 𝑘 → (𝑥 MonoAP 𝑓𝑘 MonoAP 𝑓))
5756rexralbidv 3226 . . . . . . 7 (𝑥 = 𝑘 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓))
5857ralbidv 3127 . . . . . 6 (𝑥 = 𝑘 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓))
59 breq1 5040 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (𝑥 MonoAP 𝑓 ↔ (𝑘 + 1) MonoAP 𝑓))
6059rexralbidv 3226 . . . . . . 7 (𝑥 = (𝑘 + 1) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
6160ralbidv 3127 . . . . . 6 (𝑥 = (𝑘 + 1) → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
62 breq1 5040 . . . . . . . 8 (𝑥 = 𝐾 → (𝑥 MonoAP 𝑓𝐾 MonoAP 𝑓))
6362rexralbidv 3226 . . . . . . 7 (𝑥 = 𝐾 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓))
6463ralbidv 3127 . . . . . 6 (𝑥 = 𝐾 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓))
65 hashcl 13781 . . . . . . . . 9 (𝑟 ∈ Fin → (♯‘𝑟) ∈ ℕ0)
66 nn0p1nn 11987 . . . . . . . . 9 ((♯‘𝑟) ∈ ℕ0 → ((♯‘𝑟) + 1) ∈ ℕ)
6765, 66syl 17 . . . . . . . 8 (𝑟 ∈ Fin → ((♯‘𝑟) + 1) ∈ ℕ)
68 simpll 766 . . . . . . . . . . 11 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑟 ∈ Fin)
69 simplr 768 . . . . . . . . . . . 12 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1))))
70 vex 3414 . . . . . . . . . . . . 13 𝑟 ∈ V
71 ovex 7190 . . . . . . . . . . . . 13 (1...((♯‘𝑟) + 1)) ∈ V
7270, 71elmap 8467 . . . . . . . . . . . 12 (𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1))) ↔ 𝑓:(1...((♯‘𝑟) + 1))⟶𝑟)
7369, 72sylib 221 . . . . . . . . . . 11 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓:(1...((♯‘𝑟) + 1))⟶𝑟)
74 simpr 488 . . . . . . . . . . 11 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → ¬ 2 MonoAP 𝑓)
7568, 73, 74vdwlem12 16398 . . . . . . . . . 10 ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓)
76 iman 405 . . . . . . . . . 10 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) → 2 MonoAP 𝑓) ↔ ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓))
7775, 76mpbir 234 . . . . . . . . 9 ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) → 2 MonoAP 𝑓)
7877ralrimiva 3114 . . . . . . . 8 (𝑟 ∈ Fin → ∀𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓)
79 oveq2 7165 . . . . . . . . . . 11 (𝑛 = ((♯‘𝑟) + 1) → (1...𝑛) = (1...((♯‘𝑟) + 1)))
8079oveq2d 7173 . . . . . . . . . 10 (𝑛 = ((♯‘𝑟) + 1) → (𝑟m (1...𝑛)) = (𝑟m (1...((♯‘𝑟) + 1))))
8180raleqdv 3330 . . . . . . . . 9 (𝑛 = ((♯‘𝑟) + 1) → (∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓))
8281rspcev 3544 . . . . . . . 8 ((((♯‘𝑟) + 1) ∈ ℕ ∧ ∀𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓)
8367, 78, 82syl2anc 587 . . . . . . 7 (𝑟 ∈ Fin → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓)
8483rgen 3081 . . . . . 6 𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓
85 oveq1 7164 . . . . . . . . . . 11 (𝑟 = 𝑠 → (𝑟m (1...𝑛)) = (𝑠m (1...𝑛)))
8685raleqdv 3330 . . . . . . . . . 10 (𝑟 = 𝑠 → (∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓))
8786rexbidv 3222 . . . . . . . . 9 (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓))
88 oveq2 7165 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
8988oveq2d 7173 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝑠m (1...𝑛)) = (𝑠m (1...𝑚)))
9089raleqdv 3330 . . . . . . . . . . 11 (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑓))
91 breq2 5041 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑘 MonoAP 𝑓𝑘 MonoAP 𝑔))
9291cbvralvw 3362 . . . . . . . . . . 11 (∀𝑓 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔)
9390, 92bitrdi 290 . . . . . . . . . 10 (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔))
9493cbvrexvw 3363 . . . . . . . . 9 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔)
9587, 94bitrdi 290 . . . . . . . 8 (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔))
9695cbvralvw 3362 . . . . . . 7 (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔)
97 simplr 768 . . . . . . . . . 10 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔) → 𝑟 ∈ Fin)
98 simpll 766 . . . . . . . . . 10 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔) → 𝑘 ∈ (ℤ‘2))
99 simpr 488 . . . . . . . . . . 11 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔) → ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔)
10094ralbii 3098 . . . . . . . . . . 11 (∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔)
10199, 100sylibr 237 . . . . . . . . . 10 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔) → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓)
10297, 98, 101vdwlem11 16397 . . . . . . . . 9 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓)
103102ex 416 . . . . . . . 8 ((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) → (∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
104103ralrimdva 3119 . . . . . . 7 (𝑘 ∈ (ℤ‘2) → (∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔 → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
10596, 104syl5bi 245 . . . . . 6 (𝑘 ∈ (ℤ‘2) → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓 → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
10655, 58, 61, 64, 84, 105uzind4i 12364 . . . . 5 (𝐾 ∈ (ℤ‘2) → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓)
107 oveq1 7164 . . . . . . . 8 (𝑟 = 𝑅 → (𝑟m (1...𝑛)) = (𝑅m (1...𝑛)))
108107raleqdv 3330 . . . . . . 7 (𝑟 = 𝑅 → (∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
109108rexbidv 3222 . . . . . 6 (𝑟 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
110109rspcv 3539 . . . . 5 (𝑅 ∈ Fin → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
1112, 106, 110syl2im 40 . . . 4 (𝜑 → (𝐾 ∈ (ℤ‘2) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
11252, 111jaod 856 . . 3 (𝜑 → ((𝐾 = 1 ∨ 𝐾 ∈ (ℤ‘2)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
1131, 112syl5bi 245 . 2 (𝜑 → (𝐾 ∈ ℕ → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
114 fveq2 6664 . . . . . . 7 (𝐾 = 0 → (AP‘𝐾) = (AP‘0))
115114oveqd 7174 . . . . . 6 (𝐾 = 0 → (1(AP‘𝐾)1) = (1(AP‘0)1))
116 vdwap0 16382 . . . . . . 7 ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘0)1) = ∅)
1177, 7, 116mp2an 691 . . . . . 6 (1(AP‘0)1) = ∅
118115, 117eqtrdi 2810 . . . . 5 (𝐾 = 0 → (1(AP‘𝐾)1) = ∅)
119 0ss 4296 . . . . 5 ∅ ⊆ (𝑓 “ {(𝑓‘1)})
120118, 119eqsstrdi 3949 . . . 4 (𝐾 = 0 → (1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}))
121120ralrimivw 3115 . . 3 (𝐾 = 0 → ∀𝑓 ∈ (𝑅m (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}))
122121, 51syl5 34 . 2 (𝜑 → (𝐾 = 0 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
123 elnn0 11950 . . 3 (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
12441, 123sylib 221 . 2 (𝜑 → (𝐾 ∈ ℕ ∨ 𝐾 = 0))
125113, 122, 124mpjaod 857 1 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1539  ∃wex 1782   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  Vcvv 3410   ⊆ wss 3861  ∅c0 4228  {csn 4526   class class class wbr 5037  ◡ccnv 5528   “ cima 5532   Fn wfn 6336  ⟶wf 6337  ‘cfv 6341  (class class class)co 7157   ↑m cmap 8423  Fincfn 8541  0cc0 10589  1c1 10590   + caddc 10592  ℕcn 11688  2c2 11743  ℕ0cn0 11948  ℤcz 12034  ℤ≥cuz 12296  ...cfz 12953  ♯chash 13754  APcvdwa 16371   MonoAP cvdwm 16372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466  ax-cnex 10645  ax-resscn 10646  ax-1cn 10647  ax-icn 10648  ax-addcl 10649  ax-addrcl 10650  ax-mulcl 10651  ax-mulrcl 10652  ax-mulcom 10653  ax-addass 10654  ax-mulass 10655  ax-distr 10656  ax-i2m1 10657  ax-1ne0 10658  ax-1rid 10659  ax-rnegex 10660  ax-rrecex 10661  ax-cnre 10662  ax-pre-lttri 10663  ax-pre-lttrn 10664  ax-pre-ltadd 10665  ax-pre-mulgt0 10666 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-riota 7115  df-ov 7160  df-oprab 7161  df-mpo 7162  df-om 7587  df-1st 7700  df-2nd 7701  df-wrecs 7964  df-recs 8025  df-rdg 8063  df-1o 8119  df-oadd 8123  df-er 8306  df-map 8425  df-pm 8426  df-en 8542  df-dom 8543  df-sdom 8544  df-fin 8545  df-dju 9377  df-card 9415  df-pnf 10729  df-mnf 10730  df-xr 10731  df-ltxr 10732  df-le 10733  df-sub 10924  df-neg 10925  df-nn 11689  df-2 11751  df-n0 11949  df-xnn0 12021  df-z 12035  df-uz 12297  df-rp 12445  df-fz 12954  df-hash 13755  df-vdwap 16374  df-vdwmc 16375  df-vdwpc 16376 This theorem is referenced by:  vdw  16400
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