Theorem vdwlem13 16865

Description: Lemma for vdw 16866. 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 12850	. . 3 ⊢ (𝐾 ∈ ℕ ↔ (𝐾 = 1 ∨ 𝐾 ∈ (ℤ≥‘2)))
2		vdw.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Fin)
3		ovex 7390	. . . . . . . . . 10 ⊢ (1...1) ∈ V
4		elmapg 8778	. . . . . . . . . 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 12164	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
8		vdwap1 16849	. . . . . . . . . 10 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘1)1) = {1})
9	7, 7, 8	mp2an 690	. . . . . . . . 9 ⊢ (1(AP‘1)1) = {1}
10		1z 12533	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
11		elfz3 13451	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ (1...1))
12	10, 11	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 1 ∈ (1...1))
13		eqidd 2737	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (𝑓‘1) = (𝑓‘1))
14		ffn 6668	. . . . . . . . . . . . 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 7010	. . . . . . . . . . . 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 4769	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → {1} ⊆ (◡𝑓 “ {(𝑓‘1)}))
20	9, 19	eqsstrid 3992	. . . . . . . 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 3143	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ (𝑅 ↑m (1...1))(1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
23		fveq2 6842	. . . . . . . . 9 ⊢ (𝐾 = 1 → (AP‘𝐾) = (AP‘1))
24	23	oveqd 7374	. . . . . . . 8 ⊢ (𝐾 = 1 → (1(AP‘𝐾)1) = (1(AP‘1)1))
25	24	sseq1d 3975	. . . . . . 7 ⊢ (𝐾 = 1 → ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
26	25	ralbidv 3174	. . . . . 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 7364	. . . . . . . . . . . 12 ⊢ (𝑎 = 1 → (𝑎(AP‘𝐾)𝑑) = (1(AP‘𝐾)𝑑))
29	28	sseq1d 3975	. . . . . . . . . . 11 ⊢ (𝑎 = 1 → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)})))
30		oveq2 7365	. . . . . . . . . . . 12 ⊢ (𝑑 = 1 → (1(AP‘𝐾)𝑑) = (1(AP‘𝐾)1))
31	30	sseq1d 3975	. . . . . . . . . . 11 ⊢ (𝑑 = 1 → ((1(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
32	29, 31	rspc2ev 3592	. . . . . . . . . 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 6855	. . . . . . . . . 10 ⊢ (𝑓‘1) ∈ V
35		sneq 4596	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑓‘1) → {𝑐} = {(𝑓‘1)})
36	35	imaeq2d 6013	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑓‘1) → (◡𝑓 “ {𝑐}) = (◡𝑓 “ {(𝑓‘1)}))
37	36	sseq2d 3976	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑓‘1) → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)})))
38	37	2rexbidv 3213	. . . . . . . . . 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 3565	. . . . . . . . 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 16850	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑m (1...1))) → (𝐾 MonoAP 𝑓 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐})))
44	40, 43	syl5ibr 245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑m (1...1))) → ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → 𝐾 MonoAP 𝑓))
45	44	ralimdva 3164	. . . . . 6 ⊢ (𝜑 → (∀𝑓 ∈ (𝑅 ↑m (1...1))(1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → ∀𝑓 ∈ (𝑅 ↑m (1...1))𝐾 MonoAP 𝑓))
46		oveq2 7365	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
47	46	oveq2d 7373	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑅 ↑m (1...𝑛)) = (𝑅 ↑m (1...1)))
48	47	raleqdv 3313	. . . . . . . 8 ⊢ (𝑛 = 1 → (∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑m (1...1))𝐾 MonoAP 𝑓))
49	48	rspcev 3581	. . . . . . 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 5108	. . . . . . . 8 ⊢ (𝑥 = 2 → (𝑥 MonoAP 𝑓 ↔ 2 MonoAP 𝑓))
54	53	rexralbidv 3214	. . . . . . 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 3174	. . . . . 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 5108	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝑥 MonoAP 𝑓 ↔ 𝑘 MonoAP 𝑓))
57	56	rexralbidv 3214	. . . . . . 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 3174	. . . . . 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 5108	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 MonoAP 𝑓 ↔ (𝑘 + 1) MonoAP 𝑓))
60	59	rexralbidv 3214	. . . . . . 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 3174	. . . . . 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 5108	. . . . . . . 8 ⊢ (𝑥 = 𝐾 → (𝑥 MonoAP 𝑓 ↔ 𝐾 MonoAP 𝑓))
63	62	rexralbidv 3214	. . . . . . 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 3174	. . . . . 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 14256	. . . . . . . . 9 ⊢ (𝑟 ∈ Fin → (♯‘𝑟) ∈ ℕ0)
66		nn0p1nn 12452	. . . . . . . . 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 3449	. . . . . . . . . . . . 13 ⊢ 𝑟 ∈ V
71		ovex 7390	. . . . . . . . . . . . 13 ⊢ (1...((♯‘𝑟) + 1)) ∈ V
72	70, 71	elmap 8809	. . . . . . . . . . . 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 16864	. . . . . . . . . 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 3143	. . . . . . . 8 ⊢ (𝑟 ∈ Fin → ∀𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓)
79		oveq2 7365	. . . . . . . . . . 11 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (1...𝑛) = (1...((♯‘𝑟) + 1)))
80	79	oveq2d 7373	. . . . . . . . . 10 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (𝑟 ↑m (1...𝑛)) = (𝑟 ↑m (1...((♯‘𝑟) + 1))))
81	80	raleqdv 3313	. . . . . . . . 9 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (∀𝑓 ∈ (𝑟 ↑m (1...𝑛))2 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓))
82	81	rspcev 3581	. . . . . . . 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 3066	. . . . . 6 ⊢ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑m (1...𝑛))2 MonoAP 𝑓
85		oveq1 7364	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → (𝑟 ↑m (1...𝑛)) = (𝑠 ↑m (1...𝑛)))
86	85	raleqdv 3313	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (∀𝑓 ∈ (𝑟 ↑m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑m (1...𝑛))𝑘 MonoAP 𝑓))
87	86	rexbidv 3175	. . . . . . . . 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 7365	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
89	88	oveq2d 7373	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝑠 ↑m (1...𝑛)) = (𝑠 ↑m (1...𝑚)))
90	89	raleqdv 3313	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠 ↑m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑓))
91		breq2 5109	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑘 MonoAP 𝑓 ↔ 𝑘 MonoAP 𝑔))
92	91	cbvralvw 3225	. . . . . . . . . . 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 3226	. . . . . . . . 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 3225	. . . . . . 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 3096	. . . . . . . . . . 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 16863	. . . . . . . . 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 3151	. . . . . . 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 12835	. . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘2) → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑m (1...𝑛))𝐾 MonoAP 𝑓)
107		oveq1 7364	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 ↑m (1...𝑛)) = (𝑅 ↑m (1...𝑛)))
108	107	raleqdv 3313	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∀𝑓 ∈ (𝑟 ↑m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓))
109	108	rexbidv 3175	. . . . . 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 3577	. . . . 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 6842	. . . . . . 7 ⊢ (𝐾 = 0 → (AP‘𝐾) = (AP‘0))
115	114	oveqd 7374	. . . . . 6 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) = (1(AP‘0)1))
116		vdwap0 16848	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘0)1) = ∅)
117	7, 7, 116	mp2an 690	. . . . . 6 ⊢ (1(AP‘0)1) = ∅
118	115, 117	eqtrdi 2792	. . . . 5 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) = ∅)
119		0ss 4356	. . . . 5 ⊢ ∅ ⊆ (◡𝑓 “ {(𝑓‘1)})
120	118, 119	eqsstrdi 3998	. . . 4 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
121	120	ralrimivw 3147	. . 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 12415	. . 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)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  {csn 4586   class class class wbr 5105  ^◡ccnv 5632   “ cima 5636   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ↑_m cmap 8765  Fincfn 8883  0cc0 11051  1c1 11052   + caddc 11054  ℕcn 12153  2c2 12208  ℕ₀cn0 12413  ℤcz 12499  ℤ_≥cuz 12763  ...cfz 13424  ♯chash 14230  APcvdwa 16837   MonoAP cvdwm 16838

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-hash 14231  df-vdwap 16840  df-vdwmc 16841  df-vdwpc 16842

This theorem is referenced by:  vdw  16866