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Theorem vdwlem13 15628
Description: Lemma for vdw 15629. 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 11716 . . 3 (𝐾 ∈ ℕ ↔ (𝐾 = 1 ∨ 𝐾 ∈ (ℤ‘2)))
2 vdw.r . . . . . . . . . 10 (𝜑𝑅 ∈ Fin)
3 ovex 6638 . . . . . . . . . 10 (1...1) ∈ V
4 elmapg 7822 . . . . . . . . . 10 ((𝑅 ∈ Fin ∧ (1...1) ∈ V) → (𝑓 ∈ (𝑅𝑚 (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
52, 3, 4sylancl 693 . . . . . . . . 9 (𝜑 → (𝑓 ∈ (𝑅𝑚 (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
65biimpa 501 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑅𝑚 (1...1))) → 𝑓:(1...1)⟶𝑅)
7 1nn 10982 . . . . . . . . . 10 1 ∈ ℕ
8 vdwap1 15612 . . . . . . . . . 10 ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘1)1) = {1})
97, 7, 8mp2an 707 . . . . . . . . 9 (1(AP‘1)1) = {1}
10 1z 11358 . . . . . . . . . . . 12 1 ∈ ℤ
11 elfz3 12300 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ (1...1))
1210, 11mp1i 13 . . . . . . . . . . 11 ((𝜑𝑓:(1...1)⟶𝑅) → 1 ∈ (1...1))
13 eqidd 2622 . . . . . . . . . . 11 ((𝜑𝑓:(1...1)⟶𝑅) → (𝑓‘1) = (𝑓‘1))
14 ffn 6007 . . . . . . . . . . . . 13 (𝑓:(1...1)⟶𝑅𝑓 Fn (1...1))
1514adantl 482 . . . . . . . . . . . 12 ((𝜑𝑓:(1...1)⟶𝑅) → 𝑓 Fn (1...1))
16 fniniseg 6299 . . . . . . . . . . . 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 956 . . . . . . . . . 10 ((𝜑𝑓:(1...1)⟶𝑅) → 1 ∈ (𝑓 “ {(𝑓‘1)}))
1918snssd 4314 . . . . . . . . 9 ((𝜑𝑓:(1...1)⟶𝑅) → {1} ⊆ (𝑓 “ {(𝑓‘1)}))
209, 19syl5eqss 3633 . . . . . . . 8 ((𝜑𝑓:(1...1)⟶𝑅) → (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
216, 20syldan 487 . . . . . . 7 ((𝜑𝑓 ∈ (𝑅𝑚 (1...1))) → (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
2221ralrimiva 2961 . . . . . 6 (𝜑 → ∀𝑓 ∈ (𝑅𝑚 (1...1))(1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
23 fveq2 6153 . . . . . . . . 9 (𝐾 = 1 → (AP‘𝐾) = (AP‘1))
2423oveqd 6627 . . . . . . . 8 (𝐾 = 1 → (1(AP‘𝐾)1) = (1(AP‘1)1))
2524sseq1d 3616 . . . . . . 7 (𝐾 = 1 → ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)})))
2625ralbidv 2981 . . . . . 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 237 . . . . 5 (𝜑 → (𝐾 = 1 → ∀𝑓 ∈ (𝑅𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)})))
28 oveq1 6617 . . . . . . . . . . . 12 (𝑎 = 1 → (𝑎(AP‘𝐾)𝑑) = (1(AP‘𝐾)𝑑))
2928sseq1d 3616 . . . . . . . . . . 11 (𝑎 = 1 → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)})))
30 oveq2 6618 . . . . . . . . . . . 12 (𝑑 = 1 → (1(AP‘𝐾)𝑑) = (1(AP‘𝐾)1))
3130sseq1d 3616 . . . . . . . . . . 11 (𝑑 = 1 → ((1(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)})))
3229, 31rspc2ev 3312 . . . . . . . . . 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 1411 . . . . . . . . 9 ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}))
34 fvex 6163 . . . . . . . . . 10 (𝑓‘1) ∈ V
35 sneq 4163 . . . . . . . . . . . . 13 (𝑐 = (𝑓‘1) → {𝑐} = {(𝑓‘1)})
3635imaeq2d 5430 . . . . . . . . . . . 12 (𝑐 = (𝑓‘1) → (𝑓 “ {𝑐}) = (𝑓 “ {(𝑓‘1)}))
3736sseq2d 3617 . . . . . . . . . . 11 (𝑐 = (𝑓‘1) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)})))
38372rexbidv 3051 . . . . . . . . . 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 3289 . . . . . . . . 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 481 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝑅𝑚 (1...1))) → 𝐾 ∈ ℕ0)
433, 42, 6vdwmc 15613 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑅𝑚 (1...1))) → (𝐾 MonoAP 𝑓 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐})))
4440, 43syl5ibr 236 . . . . . . 7 ((𝜑𝑓 ∈ (𝑅𝑚 (1...1))) → ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → 𝐾 MonoAP 𝑓))
4544ralimdva 2957 . . . . . 6 (𝜑 → (∀𝑓 ∈ (𝑅𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∀𝑓 ∈ (𝑅𝑚 (1...1))𝐾 MonoAP 𝑓))
46 oveq2 6618 . . . . . . . . . 10 (𝑛 = 1 → (1...𝑛) = (1...1))
4746oveq2d 6626 . . . . . . . . 9 (𝑛 = 1 → (𝑅𝑚 (1...𝑛)) = (𝑅𝑚 (1...1)))
4847raleqdv 3136 . . . . . . . 8 (𝑛 = 1 → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅𝑚 (1...1))𝐾 MonoAP 𝑓))
4948rspcev 3298 . . . . . . 7 ((1 ∈ ℕ ∧ ∀𝑓 ∈ (𝑅𝑚 (1...1))𝐾 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
507, 49mpan 705 . . . . . 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 4621 . . . . . . . 8 (𝑥 = 2 → (𝑥 MonoAP 𝑓 ↔ 2 MonoAP 𝑓))
5453rexralbidv 3052 . . . . . . 7 (𝑥 = 2 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))2 MonoAP 𝑓))
5554ralbidv 2981 . . . . . 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 4621 . . . . . . . 8 (𝑥 = 𝑘 → (𝑥 MonoAP 𝑓𝑘 MonoAP 𝑓))
5756rexralbidv 3052 . . . . . . 7 (𝑥 = 𝑘 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑘 MonoAP 𝑓))
5857ralbidv 2981 . . . . . 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 4621 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (𝑥 MonoAP 𝑓 ↔ (𝑘 + 1) MonoAP 𝑓))
6059rexralbidv 3052 . . . . . . 7 (𝑥 = (𝑘 + 1) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
6160ralbidv 2981 . . . . . 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 4621 . . . . . . . 8 (𝑥 = 𝐾 → (𝑥 MonoAP 𝑓𝐾 MonoAP 𝑓))
6362rexralbidv 3052 . . . . . . 7 (𝑥 = 𝐾 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
6463ralbidv 2981 . . . . . 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 13094 . . . . . . . . . 10 (𝑟 ∈ Fin → (#‘𝑟) ∈ ℕ0)
66 nn0p1nn 11283 . . . . . . . . . 10 ((#‘𝑟) ∈ ℕ0 → ((#‘𝑟) + 1) ∈ ℕ)
6765, 66syl 17 . . . . . . . . 9 (𝑟 ∈ Fin → ((#‘𝑟) + 1) ∈ ℕ)
68 simpll 789 . . . . . . . . . . . 12 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑟 ∈ Fin)
69 simplr 791 . . . . . . . . . . . . 13 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1))))
70 vex 3192 . . . . . . . . . . . . . 14 𝑟 ∈ V
71 ovex 6638 . . . . . . . . . . . . . 14 (1...((#‘𝑟) + 1)) ∈ V
7270, 71elmap 7837 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1))) ↔ 𝑓:(1...((#‘𝑟) + 1))⟶𝑟)
7369, 72sylib 208 . . . . . . . . . . . 12 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓:(1...((#‘𝑟) + 1))⟶𝑟)
74 simpr 477 . . . . . . . . . . . 12 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → ¬ 2 MonoAP 𝑓)
7568, 73, 74vdwlem12 15627 . . . . . . . . . . 11 ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓)
76 iman 440 . . . . . . . . . . 11 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) → 2 MonoAP 𝑓) ↔ ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓))
7775, 76mpbir 221 . . . . . . . . . 10 ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) → 2 MonoAP 𝑓)
7877ralrimiva 2961 . . . . . . . . 9 (𝑟 ∈ Fin → ∀𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))2 MonoAP 𝑓)
79 oveq2 6618 . . . . . . . . . . . 12 (𝑛 = ((#‘𝑟) + 1) → (1...𝑛) = (1...((#‘𝑟) + 1)))
8079oveq2d 6626 . . . . . . . . . . 11 (𝑛 = ((#‘𝑟) + 1) → (𝑟𝑚 (1...𝑛)) = (𝑟𝑚 (1...((#‘𝑟) + 1))))
8180raleqdv 3136 . . . . . . . . . 10 (𝑛 = ((#‘𝑟) + 1) → (∀𝑓 ∈ (𝑟𝑚 (1...𝑛))2 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))2 MonoAP 𝑓))
8281rspcev 3298 . . . . . . . . 9 ((((#‘𝑟) + 1) ∈ ℕ ∧ ∀𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))2 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))2 MonoAP 𝑓)
8367, 78, 82syl2anc 692 . . . . . . . 8 (𝑟 ∈ Fin → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))2 MonoAP 𝑓)
8483rgen 2917 . . . . . . 7 𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))2 MonoAP 𝑓
8584a1i 11 . . . . . 6 (2 ∈ ℤ → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))2 MonoAP 𝑓)
86 oveq1 6617 . . . . . . . . . . 11 (𝑟 = 𝑠 → (𝑟𝑚 (1...𝑛)) = (𝑠𝑚 (1...𝑛)))
8786raleqdv 3136 . . . . . . . . . 10 (𝑟 = 𝑠 → (∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝑘 MonoAP 𝑓))
8887rexbidv 3046 . . . . . . . . 9 (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝑘 MonoAP 𝑓))
89 oveq2 6618 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
9089oveq2d 6626 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝑠𝑚 (1...𝑛)) = (𝑠𝑚 (1...𝑚)))
9190raleqdv 3136 . . . . . . . . . . 11 (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑓))
92 breq2 4622 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑘 MonoAP 𝑓𝑘 MonoAP 𝑔))
9392cbvralv 3162 . . . . . . . . . . 11 (∀𝑓 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔)
9491, 93syl6bb 276 . . . . . . . . . 10 (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔))
9594cbvrexv 3163 . . . . . . . . 9 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔)
9688, 95syl6bb 276 . . . . . . . 8 (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔))
9796cbvralv 3162 . . . . . . 7 (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔)
98 simplr 791 . . . . . . . . . 10 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → 𝑟 ∈ Fin)
99 simpll 789 . . . . . . . . . 10 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → 𝑘 ∈ (ℤ‘2))
100 simpr 477 . . . . . . . . . . 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)
10195ralbii 2975 . . . . . . . . . . 11 (∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔)
102100, 101sylibr 224 . . . . . . . . . 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)
10398, 99, 102vdwlem11 15626 . . . . . . . . 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)
104103ex 450 . . . . . . . 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))
105104ralrimdva 2964 . . . . . . 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))
10697, 105syl5bi 232 . . . . . 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))
10755, 58, 61, 64, 85, 106uzind4 11697 . . . . 5 (𝐾 ∈ (ℤ‘2) → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
108 oveq1 6617 . . . . . . . 8 (𝑟 = 𝑅 → (𝑟𝑚 (1...𝑛)) = (𝑅𝑚 (1...𝑛)))
109108raleqdv 3136 . . . . . . 7 (𝑟 = 𝑅 → (∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
110109rexbidv 3046 . . . . . 6 (𝑟 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
111110rspcv 3294 . . . . 5 (𝑅 ∈ Fin → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
1122, 107, 111syl2im 40 . . . 4 (𝜑 → (𝐾 ∈ (ℤ‘2) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
11352, 112jaod 395 . . 3 (𝜑 → ((𝐾 = 1 ∨ 𝐾 ∈ (ℤ‘2)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
1141, 113syl5bi 232 . 2 (𝜑 → (𝐾 ∈ ℕ → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
115 fveq2 6153 . . . . . . 7 (𝐾 = 0 → (AP‘𝐾) = (AP‘0))
116115oveqd 6627 . . . . . 6 (𝐾 = 0 → (1(AP‘𝐾)1) = (1(AP‘0)1))
117 vdwap0 15611 . . . . . . 7 ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘0)1) = ∅)
1187, 7, 117mp2an 707 . . . . . 6 (1(AP‘0)1) = ∅
119116, 118syl6eq 2671 . . . . 5 (𝐾 = 0 → (1(AP‘𝐾)1) = ∅)
120 0ss 3949 . . . . 5 ∅ ⊆ (𝑓 “ {(𝑓‘1)})
121119, 120syl6eqss 3639 . . . 4 (𝐾 = 0 → (1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}))
122121ralrimivw 2962 . . 3 (𝐾 = 0 → ∀𝑓 ∈ (𝑅𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}))
123122, 51syl5 34 . 2 (𝜑 → (𝐾 = 0 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
124 elnn0 11245 . . 3 (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
12541, 124sylib 208 . 2 (𝜑 → (𝐾 ∈ ℕ ∨ 𝐾 = 0))
126114, 123, 125mpjaod 396 1 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  Vcvv 3189  wss 3559  c0 3896  {csn 4153   class class class wbr 4618  ccnv 5078  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  Fincfn 7906  0cc0 9887  1c1 9888   + caddc 9890  cn 10971  2c2 11021  0cn0 11243  cz 11328  cuz 11638  ...cfz 12275  #chash 13064  APcvdwa 15600   MonoAP cvdwm 15601
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 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
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-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-n0 11244  df-xnn0 11315  df-z 11329  df-uz 11639  df-rp 11784  df-fz 12276  df-hash 13065  df-vdwap 15603  df-vdwmc 15604  df-vdwpc 15605
This theorem is referenced by:  vdw  15629
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