vdwlem13 - Metamath Proof Explorer

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Theorem vdwlem13 16922

Description: Lemma for vdw 16923. Main induction on 𝐾; 𝐾 = 0, 𝐾 = 1 base cases. (Contributed by Mario Carneiro, 18-Aug-2014.)

**Hypotheses**
Ref	Expression
vdw.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdw.k	⊢ (𝜑 → 𝐾 ∈ ℕ₀)

**Assertion**
Ref	Expression
vdwlem13	⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (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.

Step	Hyp	Ref	Expression
1		elnn1uz2 12905	. . 3 ⊢ (𝐾 ∈ ℕ ↔ (𝐾 = 1 ∨ 𝐾 ∈ (ℤ_≥‘2)))
2		vdw.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Fin)
3		ovex 7438	. . . . . . . . . 10 ⊢ (1...1) ∈ V
4		elmapg 8829	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Fin ∧ (1...1) ∈ V) → (𝑓 ∈ (𝑅 ↑_m (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
5	2, 3, 4	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → (𝑓 ∈ (𝑅 ↑_m (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
6	5	biimpa 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑_m (1...1))) → 𝑓:(1...1)⟶𝑅)
7		1nn 12219	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
8		vdwap1 16906	. . . . . . . . . 10 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘1)1) = {1})
9	7, 7, 8	mp2an 690	. . . . . . . . 9 ⊢ (1(AP‘1)1) = {1}
10		1z 12588	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
11		elfz3 13507	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ (1...1))
12	10, 11	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 1 ∈ (1...1))
13		eqidd 2733	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (𝑓‘1) = (𝑓‘1))
14		ffn 6714	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...1)⟶𝑅 → 𝑓 Fn (1...1))
15	14	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 𝑓 Fn (1...1))
16		fniniseg 7058	. . . . . . . . . . . 12 ⊢ (𝑓 Fn (1...1) → (1 ∈ (^◡𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧ (𝑓‘1) = (𝑓‘1))))
17	15, 16	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (1 ∈ (^◡𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧ (𝑓‘1) = (𝑓‘1))))
18	12, 13, 17	mpbir2and 711	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 1 ∈ (^◡𝑓 “ {(𝑓‘1)}))
19	18	snssd 4811	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → {1} ⊆ (^◡𝑓 “ {(𝑓‘1)}))
20	9, 19	eqsstrid 4029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (1(AP‘1)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}))
21	6, 20	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑_m (1...1))) → (1(AP‘1)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}))
22	21	ralrimiva 3146	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ (𝑅 ↑_m (1...1))(1(AP‘1)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}))
23		fveq2 6888	. . . . . . . . 9 ⊢ (𝐾 = 1 → (AP‘𝐾) = (AP‘1))
24	23	oveqd 7422	. . . . . . . 8 ⊢ (𝐾 = 1 → (1(AP‘𝐾)1) = (1(AP‘1)1))
25	24	sseq1d 4012	. . . . . . 7 ⊢ (𝐾 = 1 → ((1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘1)1) ⊆ (^◡𝑓 “ {(𝑓‘1)})))
26	25	ralbidv 3177	. . . . . 6 ⊢ (𝐾 = 1 → (∀𝑓 ∈ (𝑅 ↑_m (1...1))(1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}) ↔ ∀𝑓 ∈ (𝑅 ↑_m (1...1))(1(AP‘1)1) ⊆ (^◡𝑓 “ {(𝑓‘1)})))
27	22, 26	syl5ibrcom 246	. . . . 5 ⊢ (𝜑 → (𝐾 = 1 → ∀𝑓 ∈ (𝑅 ↑_m (1...1))(1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)})))
28		oveq1 7412	. . . . . . . . . . . 12 ⊢ (𝑎 = 1 → (𝑎(AP‘𝐾)𝑑) = (1(AP‘𝐾)𝑑))
29	28	sseq1d 4012	. . . . . . . . . . 11 ⊢ (𝑎 = 1 → ((𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)𝑑) ⊆ (^◡𝑓 “ {(𝑓‘1)})))
30		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑑 = 1 → (1(AP‘𝐾)𝑑) = (1(AP‘𝐾)1))
31	30	sseq1d 4012	. . . . . . . . . . 11 ⊢ (𝑑 = 1 → ((1(AP‘𝐾)𝑑) ⊆ (^◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)})))
32	29, 31	rspc2ev 3623	. . . . . . . . . 10 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ ∧ (1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝑓 “ {(𝑓‘1)}))
33	7, 7, 32	mp3an12 1451	. . . . . . . . 9 ⊢ ((1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝑓 “ {(𝑓‘1)}))
34		fvex 6901	. . . . . . . . . 10 ⊢ (𝑓‘1) ∈ V
35		sneq 4637	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑓‘1) → {𝑐} = {(𝑓‘1)})
36	35	imaeq2d 6057	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑓‘1) → (^◡𝑓 “ {𝑐}) = (^◡𝑓 “ {(𝑓‘1)}))
37	36	sseq2d 4013	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑓‘1) → ((𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝑓 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝑓 “ {(𝑓‘1)})))
38	37	2rexbidv 3219	. . . . . . . . . 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)})))
39	34, 38	spcev 3596	. . . . . . . . 9 ⊢ (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝑓 “ {(𝑓‘1)}) → ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝑓 “ {𝑐}))
40	33, 39	syl 17	. . . . . . . 8 ⊢ ((1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}) → ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝑓 “ {𝑐}))
41		vdw.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
42	41	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑_m (1...1))) → 𝐾 ∈ ℕ₀)
43	3, 42, 6	vdwmc 16907	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑_m (1...1))) → (𝐾 MonoAP 𝑓 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝑓 “ {𝑐})))
44	40, 43	imbitrrid 245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑_m (1...1))) → ((1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}) → 𝐾 MonoAP 𝑓))
45	44	ralimdva 3167	. . . . . 6 ⊢ (𝜑 → (∀𝑓 ∈ (𝑅 ↑_m (1...1))(1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}) → ∀𝑓 ∈ (𝑅 ↑_m (1...1))𝐾 MonoAP 𝑓))
46		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
47	46	oveq2d 7421	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑅 ↑_m (1...𝑛)) = (𝑅 ↑_m (1...1)))
48	47	raleqdv 3325	. . . . . . . 8 ⊢ (𝑛 = 1 → (∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑_m (1...1))𝐾 MonoAP 𝑓))
49	48	rspcev 3612	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ ∀𝑓 ∈ (𝑅 ↑_m (1...1))𝐾 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓)
50	7, 49	mpan 688	. . . . . 6 ⊢ (∀𝑓 ∈ (𝑅 ↑_m (1...1))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓)
51	45, 50	syl6 35	. . . . 5 ⊢ (𝜑 → (∀𝑓 ∈ (𝑅 ↑_m (1...1))(1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓))
52	27, 51	syld 47	. . . 4 ⊢ (𝜑 → (𝐾 = 1 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓))
53		breq1 5150	. . . . . . . 8 ⊢ (𝑥 = 2 → (𝑥 MonoAP 𝑓 ↔ 2 MonoAP 𝑓))
54	53	rexralbidv 3220	. . . . . . 7 ⊢ (𝑥 = 2 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))2 MonoAP 𝑓))
55	54	ralbidv 3177	. . . . . 6 ⊢ (𝑥 = 2 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))2 MonoAP 𝑓))
56		breq1 5150	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝑥 MonoAP 𝑓 ↔ 𝑘 MonoAP 𝑓))
57	56	rexralbidv 3220	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑘 MonoAP 𝑓))
58	57	ralbidv 3177	. . . . . 6 ⊢ (𝑥 = 𝑘 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑘 MonoAP 𝑓))
59		breq1 5150	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 MonoAP 𝑓 ↔ (𝑘 + 1) MonoAP 𝑓))
60	59	rexralbidv 3220	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
61	60	ralbidv 3177	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
62		breq1 5150	. . . . . . . 8 ⊢ (𝑥 = 𝐾 → (𝑥 MonoAP 𝑓 ↔ 𝐾 MonoAP 𝑓))
63	62	rexralbidv 3220	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝐾 MonoAP 𝑓))
64	63	ralbidv 3177	. . . . . 6 ⊢ (𝑥 = 𝐾 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝐾 MonoAP 𝑓))
65		hashcl 14312	. . . . . . . . 9 ⊢ (𝑟 ∈ Fin → (♯‘𝑟) ∈ ℕ₀)
66		nn0p1nn 12507	. . . . . . . . 9 ⊢ ((♯‘𝑟) ∈ ℕ₀ → ((♯‘𝑟) + 1) ∈ ℕ)
67	65, 66	syl 17	. . . . . . . 8 ⊢ (𝑟 ∈ Fin → ((♯‘𝑟) + 1) ∈ ℕ)
68		simpll 765	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑟 ∈ Fin)
69		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1))))
70		vex 3478	. . . . . . . . . . . . 13 ⊢ 𝑟 ∈ V
71		ovex 7438	. . . . . . . . . . . . 13 ⊢ (1...((♯‘𝑟) + 1)) ∈ V
72	70, 71	elmap 8861	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1))) ↔ 𝑓:(1...((♯‘𝑟) + 1))⟶𝑟)
73	69, 72	sylib 217	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓:(1...((♯‘𝑟) + 1))⟶𝑟)
74		simpr 485	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → ¬ 2 MonoAP 𝑓)
75	68, 73, 74	vdwlem12 16921	. . . . . . . . . 10 ⊢ ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓)
76		iman 402	. . . . . . . . . 10 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1)))) → 2 MonoAP 𝑓) ↔ ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓))
77	75, 76	mpbir 230	. . . . . . . . 9 ⊢ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1)))) → 2 MonoAP 𝑓)
78	77	ralrimiva 3146	. . . . . . . 8 ⊢ (𝑟 ∈ Fin → ∀𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓)
79		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (1...𝑛) = (1...((♯‘𝑟) + 1)))
80	79	oveq2d 7421	. . . . . . . . . 10 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (𝑟 ↑_m (1...𝑛)) = (𝑟 ↑_m (1...((♯‘𝑟) + 1))))
81	80	raleqdv 3325	. . . . . . . . 9 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))2 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓))
82	81	rspcev 3612	. . . . . . . 8 ⊢ ((((♯‘𝑟) + 1) ∈ ℕ ∧ ∀𝑓 ∈ (𝑟 ↑_m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))2 MonoAP 𝑓)
83	67, 78, 82	syl2anc 584	. . . . . . 7 ⊢ (𝑟 ∈ Fin → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))2 MonoAP 𝑓)
84	83	rgen 3063	. . . . . 6 ⊢ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))2 MonoAP 𝑓
85		oveq1 7412	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → (𝑟 ↑_m (1...𝑛)) = (𝑠 ↑_m (1...𝑛)))
86	85	raleqdv 3325	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑_m (1...𝑛))𝑘 MonoAP 𝑓))
87	86	rexbidv 3178	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_m (1...𝑛))𝑘 MonoAP 𝑓))
88		oveq2 7413	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
89	88	oveq2d 7421	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝑠 ↑_m (1...𝑛)) = (𝑠 ↑_m (1...𝑚)))
90	89	raleqdv 3325	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠 ↑_m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑓))
91		breq2 5151	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑘 MonoAP 𝑓 ↔ 𝑘 MonoAP 𝑔))
92	91	cbvralvw 3234	. . . . . . . . . . 11 ⊢ (∀𝑓 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔)
93	90, 92	bitrdi 286	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠 ↑_m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔))
94	93	cbvrexvw 3235	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔)
95	87, 94	bitrdi 286	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔))
96	95	cbvralvw 3234	. . . . . . 7 ⊢ (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔)
97		simplr 767	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (ℤ_≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔) → 𝑟 ∈ Fin)
98		simpll 765	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (ℤ_≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔) → 𝑘 ∈ (ℤ_≥‘2))
99		simpr 485	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ (ℤ_≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔) → ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔)
100	94	ralbii 3093	. . . . . . . . . . 11 ⊢ (∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔)
101	99, 100	sylibr 233	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (ℤ_≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔) → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_m (1...𝑛))𝑘 MonoAP 𝑓)
102	97, 98, 101	vdwlem11 16920	. . . . . . . . 9 ⊢ (((𝑘 ∈ (ℤ_≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))(𝑘 + 1) MonoAP 𝑓)
103	102	ex 413	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ_≥‘2) ∧ 𝑟 ∈ Fin) → (∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
104	103	ralrimdva 3154	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘2) → (∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑_m (1...𝑚))𝑘 MonoAP 𝑔 → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
105	96, 104	biimtrid 241	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘2) → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝑘 MonoAP 𝑓 → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
106	55, 58, 61, 64, 84, 105	uzind4i 12890	. . . . 5 ⊢ (𝐾 ∈ (ℤ_≥‘2) → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝐾 MonoAP 𝑓)
107		oveq1 7412	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 ↑_m (1...𝑛)) = (𝑅 ↑_m (1...𝑛)))
108	107	raleqdv 3325	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓))
109	108	rexbidv 3178	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓))
110	109	rspcv 3608	. . . . 5 ⊢ (𝑅 ∈ Fin → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑_m (1...𝑛))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓))
111	2, 106, 110	syl2im 40	. . . 4 ⊢ (𝜑 → (𝐾 ∈ (ℤ_≥‘2) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓))
112	52, 111	jaod 857	. . 3 ⊢ (𝜑 → ((𝐾 = 1 ∨ 𝐾 ∈ (ℤ_≥‘2)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓))
113	1, 112	biimtrid 241	. 2 ⊢ (𝜑 → (𝐾 ∈ ℕ → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓))
114		fveq2 6888	. . . . . . 7 ⊢ (𝐾 = 0 → (AP‘𝐾) = (AP‘0))
115	114	oveqd 7422	. . . . . 6 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) = (1(AP‘0)1))
116		vdwap0 16905	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘0)1) = ∅)
117	7, 7, 116	mp2an 690	. . . . . 6 ⊢ (1(AP‘0)1) = ∅
118	115, 117	eqtrdi 2788	. . . . 5 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) = ∅)
119		0ss 4395	. . . . 5 ⊢ ∅ ⊆ (^◡𝑓 “ {(𝑓‘1)})
120	118, 119	eqsstrdi 4035	. . . 4 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}))
121	120	ralrimivw 3150	. . 3 ⊢ (𝐾 = 0 → ∀𝑓 ∈ (𝑅 ↑_m (1...1))(1(AP‘𝐾)1) ⊆ (^◡𝑓 “ {(𝑓‘1)}))
122	121, 51	syl5 34	. 2 ⊢ (𝜑 → (𝐾 = 0 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓))
123		elnn0 12470	. . 3 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
124	41, 123	sylib 217	. 2 ⊢ (𝜑 → (𝐾 ∈ ℕ ∨ 𝐾 = 0))
125	113, 122, 124	mpjaod 858	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))𝐾 MonoAP 𝑓)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ⊆ wss 3947 ∅c0 4321 {csn 4627 class class class wbr 5147 ^◡ccnv 5674 “ cima 5678 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 Fincfn 8935 0cc0 11106 1c1 11107 + caddc 11109 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 ♯chash 14286 APcvdwa 16894 MonoAP cvdwm 16895

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-hash 14287 df-vdwap 16897 df-vdwmc 16898 df-vdwpc 16899

This theorem is referenced by: vdw 16923

W3C validator