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Theorem vdwlem13 15535
 Description: Lemma for vdw 15536. 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 11641 . . 3 (𝐾 ∈ ℕ ↔ (𝐾 = 1 ∨ 𝐾 ∈ (ℤ‘2)))
2 vdw.r . . . . . . . . . 10 (𝜑𝑅 ∈ Fin)
3 ovex 6577 . . . . . . . . . 10 (1...1) ∈ V
4 elmapg 7757 . . . . . . . . . 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 500 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑅𝑚 (1...1))) → 𝑓:(1...1)⟶𝑅)
7 1nn 10908 . . . . . . . . . 10 1 ∈ ℕ
8 vdwap1 15519 . . . . . . . . . 10 ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘1)1) = {1})
97, 7, 8mp2an 704 . . . . . . . . 9 (1(AP‘1)1) = {1}
10 1z 11284 . . . . . . . . . . . 12 1 ∈ ℤ
11 elfz3 12222 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ (1...1))
1210, 11mp1i 13 . . . . . . . . . . 11 ((𝜑𝑓:(1...1)⟶𝑅) → 1 ∈ (1...1))
13 eqidd 2611 . . . . . . . . . . 11 ((𝜑𝑓:(1...1)⟶𝑅) → (𝑓‘1) = (𝑓‘1))
14 ffn 5958 . . . . . . . . . . . . 13 (𝑓:(1...1)⟶𝑅𝑓 Fn (1...1))
1514adantl 481 . . . . . . . . . . . 12 ((𝜑𝑓:(1...1)⟶𝑅) → 𝑓 Fn (1...1))
16 fniniseg 6246 . . . . . . . . . . . 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 959 . . . . . . . . . 10 ((𝜑𝑓:(1...1)⟶𝑅) → 1 ∈ (𝑓 “ {(𝑓‘1)}))
1918snssd 4281 . . . . . . . . 9 ((𝜑𝑓:(1...1)⟶𝑅) → {1} ⊆ (𝑓 “ {(𝑓‘1)}))
209, 19syl5eqss 3612 . . . . . . . 8 ((𝜑𝑓:(1...1)⟶𝑅) → (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
216, 20syldan 486 . . . . . . 7 ((𝜑𝑓 ∈ (𝑅𝑚 (1...1))) → (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
2221ralrimiva 2949 . . . . . 6 (𝜑 → ∀𝑓 ∈ (𝑅𝑚 (1...1))(1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
23 fveq2 6103 . . . . . . . . 9 (𝐾 = 1 → (AP‘𝐾) = (AP‘1))
2423oveqd 6566 . . . . . . . 8 (𝐾 = 1 → (1(AP‘𝐾)1) = (1(AP‘1)1))
2524sseq1d 3595 . . . . . . 7 (𝐾 = 1 → ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)})))
2625ralbidv 2969 . . . . . 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 236 . . . . 5 (𝜑 → (𝐾 = 1 → ∀𝑓 ∈ (𝑅𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)})))
28 oveq1 6556 . . . . . . . . . . . 12 (𝑎 = 1 → (𝑎(AP‘𝐾)𝑑) = (1(AP‘𝐾)𝑑))
2928sseq1d 3595 . . . . . . . . . . 11 (𝑎 = 1 → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)})))
30 oveq2 6557 . . . . . . . . . . . 12 (𝑑 = 1 → (1(AP‘𝐾)𝑑) = (1(AP‘𝐾)1))
3130sseq1d 3595 . . . . . . . . . . 11 (𝑑 = 1 → ((1(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)})))
3229, 31rspc2ev 3295 . . . . . . . . . 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 1406 . . . . . . . . 9 ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}))
34 fvex 6113 . . . . . . . . . 10 (𝑓‘1) ∈ V
35 sneq 4135 . . . . . . . . . . . . 13 (𝑐 = (𝑓‘1) → {𝑐} = {(𝑓‘1)})
3635imaeq2d 5385 . . . . . . . . . . . 12 (𝑐 = (𝑓‘1) → (𝑓 “ {𝑐}) = (𝑓 “ {(𝑓‘1)}))
3736sseq2d 3596 . . . . . . . . . . 11 (𝑐 = (𝑓‘1) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)})))
38372rexbidv 3039 . . . . . . . . . 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 3273 . . . . . . . . 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 480 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝑅𝑚 (1...1))) → 𝐾 ∈ ℕ0)
433, 42, 6vdwmc 15520 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑅𝑚 (1...1))) → (𝐾 MonoAP 𝑓 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐})))
4440, 43syl5ibr 235 . . . . . . 7 ((𝜑𝑓 ∈ (𝑅𝑚 (1...1))) → ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → 𝐾 MonoAP 𝑓))
4544ralimdva 2945 . . . . . 6 (𝜑 → (∀𝑓 ∈ (𝑅𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∀𝑓 ∈ (𝑅𝑚 (1...1))𝐾 MonoAP 𝑓))
46 oveq2 6557 . . . . . . . . . 10 (𝑛 = 1 → (1...𝑛) = (1...1))
4746oveq2d 6565 . . . . . . . . 9 (𝑛 = 1 → (𝑅𝑚 (1...𝑛)) = (𝑅𝑚 (1...1)))
4847raleqdv 3121 . . . . . . . 8 (𝑛 = 1 → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅𝑚 (1...1))𝐾 MonoAP 𝑓))
4948rspcev 3282 . . . . . . 7 ((1 ∈ ℕ ∧ ∀𝑓 ∈ (𝑅𝑚 (1...1))𝐾 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
507, 49mpan 702 . . . . . 6 (∀𝑓 ∈ (𝑅𝑚 (1...1))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
5145, 50syl6 34 . . . . 5 (𝜑 → (∀𝑓 ∈ (𝑅𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
5227, 51syld 46 . . . 4 (𝜑 → (𝐾 = 1 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
53 breq1 4586 . . . . . . . 8 (𝑥 = 2 → (𝑥 MonoAP 𝑓 ↔ 2 MonoAP 𝑓))
5453rexralbidv 3040 . . . . . . 7 (𝑥 = 2 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))2 MonoAP 𝑓))
5554ralbidv 2969 . . . . . 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 4586 . . . . . . . 8 (𝑥 = 𝑘 → (𝑥 MonoAP 𝑓𝑘 MonoAP 𝑓))
5756rexralbidv 3040 . . . . . . 7 (𝑥 = 𝑘 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑘 MonoAP 𝑓))
5857ralbidv 2969 . . . . . 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 4586 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (𝑥 MonoAP 𝑓 ↔ (𝑘 + 1) MonoAP 𝑓))
6059rexralbidv 3040 . . . . . . 7 (𝑥 = (𝑘 + 1) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
6160ralbidv 2969 . . . . . 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 4586 . . . . . . . 8 (𝑥 = 𝐾 → (𝑥 MonoAP 𝑓𝐾 MonoAP 𝑓))
6362rexralbidv 3040 . . . . . . 7 (𝑥 = 𝐾 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
6463ralbidv 2969 . . . . . 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 13009 . . . . . . . . . 10 (𝑟 ∈ Fin → (#‘𝑟) ∈ ℕ0)
66 nn0p1nn 11209 . . . . . . . . . 10 ((#‘𝑟) ∈ ℕ0 → ((#‘𝑟) + 1) ∈ ℕ)
6765, 66syl 17 . . . . . . . . 9 (𝑟 ∈ Fin → ((#‘𝑟) + 1) ∈ ℕ)
68 simpll 786 . . . . . . . . . . . 12 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑟 ∈ Fin)
69 simplr 788 . . . . . . . . . . . . 13 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1))))
70 vex 3176 . . . . . . . . . . . . . 14 𝑟 ∈ V
71 ovex 6577 . . . . . . . . . . . . . 14 (1...((#‘𝑟) + 1)) ∈ V
7270, 71elmap 7772 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1))) ↔ 𝑓:(1...((#‘𝑟) + 1))⟶𝑟)
7369, 72sylib 207 . . . . . . . . . . . 12 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓:(1...((#‘𝑟) + 1))⟶𝑟)
74 simpr 476 . . . . . . . . . . . 12 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → ¬ 2 MonoAP 𝑓)
7568, 73, 74vdwlem12 15534 . . . . . . . . . . 11 ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓)
76 iman 439 . . . . . . . . . . 11 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) → 2 MonoAP 𝑓) ↔ ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓))
7775, 76mpbir 220 . . . . . . . . . 10 ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))) → 2 MonoAP 𝑓)
7877ralrimiva 2949 . . . . . . . . 9 (𝑟 ∈ Fin → ∀𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))2 MonoAP 𝑓)
79 oveq2 6557 . . . . . . . . . . . 12 (𝑛 = ((#‘𝑟) + 1) → (1...𝑛) = (1...((#‘𝑟) + 1)))
8079oveq2d 6565 . . . . . . . . . . 11 (𝑛 = ((#‘𝑟) + 1) → (𝑟𝑚 (1...𝑛)) = (𝑟𝑚 (1...((#‘𝑟) + 1))))
8180raleqdv 3121 . . . . . . . . . 10 (𝑛 = ((#‘𝑟) + 1) → (∀𝑓 ∈ (𝑟𝑚 (1...𝑛))2 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))2 MonoAP 𝑓))
8281rspcev 3282 . . . . . . . . 9 ((((#‘𝑟) + 1) ∈ ℕ ∧ ∀𝑓 ∈ (𝑟𝑚 (1...((#‘𝑟) + 1)))2 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))2 MonoAP 𝑓)
8367, 78, 82syl2anc 691 . . . . . . . 8 (𝑟 ∈ Fin → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))2 MonoAP 𝑓)
8483rgen 2906 . . . . . . 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 6556 . . . . . . . . . . 11 (𝑟 = 𝑠 → (𝑟𝑚 (1...𝑛)) = (𝑠𝑚 (1...𝑛)))
8786raleqdv 3121 . . . . . . . . . 10 (𝑟 = 𝑠 → (∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝑘 MonoAP 𝑓))
8887rexbidv 3034 . . . . . . . . 9 (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝑘 MonoAP 𝑓))
89 oveq2 6557 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
9089oveq2d 6565 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝑠𝑚 (1...𝑛)) = (𝑠𝑚 (1...𝑚)))
9190raleqdv 3121 . . . . . . . . . . 11 (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑓))
92 breq2 4587 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑘 MonoAP 𝑓𝑘 MonoAP 𝑔))
9392cbvralv 3147 . . . . . . . . . . 11 (∀𝑓 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔)
9491, 93syl6bb 275 . . . . . . . . . 10 (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔))
9594cbvrexv 3148 . . . . . . . . 9 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔)
9688, 95syl6bb 275 . . . . . . . 8 (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔))
9796cbvralv 3147 . . . . . . 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 788 . . . . . . . . . 10 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → 𝑟 ∈ Fin)
99 simpll 786 . . . . . . . . . 10 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → 𝑘 ∈ (ℤ‘2))
100 simpr 476 . . . . . . . . . . 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 2963 . . . . . . . . . . 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 223 . . . . . . . . . 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 15533 . . . . . . . . 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 449 . . . . . . . 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 2952 . . . . . . 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 231 . . . . . 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 11622 . . . . 5 (𝐾 ∈ (ℤ‘2) → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
108 oveq1 6556 . . . . . . . 8 (𝑟 = 𝑅 → (𝑟𝑚 (1...𝑛)) = (𝑅𝑚 (1...𝑛)))
109108raleqdv 3121 . . . . . . 7 (𝑟 = 𝑅 → (∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
110109rexbidv 3034 . . . . . 6 (𝑟 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
111110rspcv 3278 . . . . 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 39 . . . 4 (𝜑 → (𝐾 ∈ (ℤ‘2) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
11352, 112jaod 394 . . 3 (𝜑 → ((𝐾 = 1 ∨ 𝐾 ∈ (ℤ‘2)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
1141, 113syl5bi 231 . 2 (𝜑 → (𝐾 ∈ ℕ → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
115 fveq2 6103 . . . . . . 7 (𝐾 = 0 → (AP‘𝐾) = (AP‘0))
116115oveqd 6566 . . . . . 6 (𝐾 = 0 → (1(AP‘𝐾)1) = (1(AP‘0)1))
117 vdwap0 15518 . . . . . . 7 ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘0)1) = ∅)
1187, 7, 117mp2an 704 . . . . . 6 (1(AP‘0)1) = ∅
119116, 118syl6eq 2660 . . . . 5 (𝐾 = 0 → (1(AP‘𝐾)1) = ∅)
120 0ss 3924 . . . . 5 ∅ ⊆ (𝑓 “ {(𝑓‘1)})
121119, 120syl6eqss 3618 . . . 4 (𝐾 = 0 → (1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}))
122121ralrimivw 2950 . . 3 (𝐾 = 0 → ∀𝑓 ∈ (𝑅𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}))
123122, 51syl5 33 . 2 (𝜑 → (𝐾 = 0 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
124 elnn0 11171 . . 3 (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
12541, 124sylib 207 . 2 (𝜑 → (𝐾 ∈ ℕ ∨ 𝐾 = 0))
126114, 123, 125mpjaod 395 1 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {csn 4125   class class class wbr 4583  ◡ccnv 5037   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Fincfn 7841  0cc0 9815  1c1 9816   + caddc 9818  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  #chash 12979  APcvdwa 15507   MonoAP cvdwm 15508 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-hash 12980  df-vdwap 15510  df-vdwmc 15511  df-vdwpc 15512 This theorem is referenced by:  vdw  15536
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