Theorem vdwlem13 17183

Description: Lemma for vdw 17184. 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 13062	. . 3 ⊢ (𝐾 ∈ ℕ ↔ (𝐾 = 1 ∨ 𝐾 ∈ (ℤ≥‘2)))
2		vdw.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Fin)
3		ovex 7460	. . . . . . . . . 10 ⊢ (1...1) ∈ V
4		elmapg 8870	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Fin ∧ (1...1) ∈ V) → (𝑓 ∈ (𝑅 ↑m (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
5	2, 3, 4	sylancl 598	. . . . . . . . 9 ⊢ (𝜑 → (𝑓 ∈ (𝑅 ↑m (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
6	5	biimpa 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑m (1...1))) → 𝑓:(1...1)⟶𝑅)
7		1nn 12356	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
8		vdwap1 17167	. . . . . . . . . 10 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘1)1) = {1})
9	7, 7, 8	mp2an 705	. . . . . . . . 9 ⊢ (1(AP‘1)1) = {1}
10		1z 12736	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
11		elfz3 13677	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ (1...1))
12	10, 11	mp1i 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 1 ∈ (1...1))
13		eqidd 2764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (𝑓‘1) = (𝑓‘1))
14		ffn 6717	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...1)⟶𝑅 → 𝑓 Fn (1...1))
15	14	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 𝑓 Fn (1...1))
16		fniniseg 7067	. . . . . . . . . . . 12 ⊢ (𝑓 Fn (1...1) → (1 ∈ (◡𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧ (𝑓‘1) = (𝑓‘1))))
17	15, 16	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (1 ∈ (◡𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧ (𝑓‘1) = (𝑓‘1))))
18	12, 13, 17	mpbir2and 726	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 1 ∈ (◡𝑓 “ {(𝑓‘1)}))
19	18	snssd 4746	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → {1} ⊆ (◡𝑓 “ {(𝑓‘1)}))
20	9, 19	eqsstrid 3950	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
21	6, 20	syldan 603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑m (1...1))) → (1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
22	21	ralrimiva 3157	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ (𝑅 ↑m (1...1))(1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
23		fveq2 6893	. . . . . . . . 9 ⊢ (𝐾 = 1 → (AP‘𝐾) = (AP‘1))
24	23	oveqd 7444	. . . . . . . 8 ⊢ (𝐾 = 1 → (1(AP‘𝐾)1) = (1(AP‘1)1))
25	24	sseq1d 3943	. . . . . . 7 ⊢ (𝐾 = 1 → ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
26	25	ralbidv 3188	. . . . . 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 250	. . . . 5 ⊢ (𝜑 → (𝐾 = 1 → ∀𝑓 ∈ (𝑅 ↑m (1...1))(1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
28		oveq1 7434	. . . . . . . . . . . 12 ⊢ (𝑎 = 1 → (𝑎(AP‘𝐾)𝑑) = (1(AP‘𝐾)𝑑))
29	28	sseq1d 3943	. . . . . . . . . . 11 ⊢ (𝑎 = 1 → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)})))
30		oveq2 7435	. . . . . . . . . . . 12 ⊢ (𝑑 = 1 → (1(AP‘𝐾)𝑑) = (1(AP‘𝐾)1))
31	30	sseq1d 3943	. . . . . . . . . . 11 ⊢ (𝑑 = 1 → ((1(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
32	29, 31	rspc2ev 3591	. . . . . . . . . 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 1480	. . . . . . . . 9 ⊢ ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}))
34		fvex 6906	. . . . . . . . . 10 ⊢ (𝑓‘1) ∈ V
35		sneq 4593	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑓‘1) → {𝑐} = {(𝑓‘1)})
36	35	imaeq2d 6057	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑓‘1) → (◡𝑓 “ {𝑐}) = (◡𝑓 “ {(𝑓‘1)}))
37	36	sseq2d 3944	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑓‘1) → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)})))
38	37	2rexbidv 3230	. . . . . . . . . 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 3563	. . . . . . . . 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 18	. . . . . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑m (1...1))) → 𝐾 ∈ ℕ0)
43	3, 42, 6	vdwmc 17168	. . . . . . . 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 249	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑m (1...1))) → ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → 𝐾 MonoAP 𝑓))
45	44	ralimdva 3177	. . . . . 6 ⊢ (𝜑 → (∀𝑓 ∈ (𝑅 ↑m (1...1))(1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → ∀𝑓 ∈ (𝑅 ↑m (1...1))𝐾 MonoAP 𝑓))
46		oveq2 7435	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
47	46	oveq2d 7443	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑅 ↑m (1...𝑛)) = (𝑅 ↑m (1...1)))
48	47	raleqdv 3322	. . . . . . . 8 ⊢ (𝑛 = 1 → (∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑m (1...1))𝐾 MonoAP 𝑓))
49	48	rspcev 3579	. . . . . . 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 703	. . . . . 6 ⊢ (∀𝑓 ∈ (𝑅 ↑m (1...1))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓)
51	45, 50	syl6 36	. . . . 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 48	. . . 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 3231	. . . . . . 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 3188	. . . . . 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 3231	. . . . . . 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 3188	. . . . . 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 3231	. . . . . . 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 3188	. . . . . 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 3231	. . . . . . 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 3188	. . . . . 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 14510	. . . . . . . . 9 ⊢ (𝑟 ∈ Fin → (♯‘𝑟) ∈ ℕ0)
66		nn0p1nn 12655	. . . . . . . . 9 ⊢ ((♯‘𝑟) ∈ ℕ0 → ((♯‘𝑟) + 1) ∈ ℕ)
67	65, 66	syl 18	. . . . . . . 8 ⊢ (𝑟 ∈ Fin → ((♯‘𝑟) + 1) ∈ ℕ)
68		simpll 779	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑟 ∈ Fin)
69		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1))))
70		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑟 ∈ V
71		ovex 7460	. . . . . . . . . . . . 13 ⊢ (1...((♯‘𝑟) + 1)) ∈ V
72	70, 71	elmap 8910	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1))) ↔ 𝑓:(1...((♯‘𝑟) + 1))⟶𝑟)
73	69, 72	sylib 221	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓:(1...((♯‘𝑟) + 1))⟶𝑟)
74		simpr 490	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → ¬ 2 MonoAP 𝑓)
75	68, 73, 74	vdwlem12 17182	. . . . . . . . . 10 ⊢ ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓)
76		iman 407	. . . . . . . . . 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 234	. . . . . . . . 9 ⊢ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) → 2 MonoAP 𝑓)
78	77	ralrimiva 3157	. . . . . . . 8 ⊢ (𝑟 ∈ Fin → ∀𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓)
79		oveq2 7435	. . . . . . . . . . 11 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (1...𝑛) = (1...((♯‘𝑟) + 1)))
80	79	oveq2d 7443	. . . . . . . . . 10 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (𝑟 ↑m (1...𝑛)) = (𝑟 ↑m (1...((♯‘𝑟) + 1))))
81	80	raleqdv 3322	. . . . . . . . 9 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (∀𝑓 ∈ (𝑟 ↑m (1...𝑛))2 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓))
82	81	rspcev 3579	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝑟 ∈ Fin → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑m (1...𝑛))2 MonoAP 𝑓)
84	83	rgen 3081	. . . . . 6 ⊢ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑m (1...𝑛))2 MonoAP 𝑓
85		oveq1 7434	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → (𝑟 ↑m (1...𝑛)) = (𝑠 ↑m (1...𝑛)))
86	85	raleqdv 3322	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (∀𝑓 ∈ (𝑟 ↑m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑m (1...𝑛))𝑘 MonoAP 𝑓))
87	86	rexbidv 3189	. . . . . . . . 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 7435	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
89	88	oveq2d 7443	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝑠 ↑m (1...𝑛)) = (𝑠 ↑m (1...𝑚)))
90	89	raleqdv 3322	. . . . . . . . . . 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 3243	. . . . . . . . . . 11 ⊢ (∀𝑓 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑔)
93	90, 92	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠 ↑m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑔))
94	93	cbvrexvw 3244	. . . . . . . . 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 290	. . . . . . . 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 3243	. . . . . . 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 781	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑔) → 𝑟 ∈ Fin)
98		simpll 779	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑔) → 𝑘 ∈ (ℤ≥‘2))
99	94	ralbii 3111	. . . . . . . . . . 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)
100	99	bilanri 512	. . . . . . . . . 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)
101	97, 98, 100	vdwlem11 17181	. . . . . . . . 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)
102	101	ex 418	. . . . . . . 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))
103	102	ralrimdva 3165	. . . . . . 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))
104	96, 103	biimtrid 245	. . . . . 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))
105	55, 58, 61, 64, 84, 104	uzind4i 13047	. . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘2) → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑m (1...𝑛))𝐾 MonoAP 𝑓)
106		oveq1 7434	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 ↑m (1...𝑛)) = (𝑅 ↑m (1...𝑛)))
107	106	raleqdv 3322	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∀𝑓 ∈ (𝑟 ↑m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓))
108	107	rexbidv 3189	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓))
109	108	rspcv 3575	. . . . 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))
110	2, 105, 109	syl2im 41	. . . 4 ⊢ (𝜑 → (𝐾 ∈ (ℤ≥‘2) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓))
111	52, 110	jaod 873	. . 3 ⊢ (𝜑 → ((𝐾 = 1 ∨ 𝐾 ∈ (ℤ≥‘2)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓))
112	1, 111	biimtrid 245	. 2 ⊢ (𝜑 → (𝐾 ∈ ℕ → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓))
113		fveq2 6893	. . . . . . 7 ⊢ (𝐾 = 0 → (AP‘𝐾) = (AP‘0))
114	113	oveqd 7444	. . . . . 6 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) = (1(AP‘0)1))
115		vdwap0 17166	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘0)1) = ∅)
116	7, 7, 115	mp2an 705	. . . . . 6 ⊢ (1(AP‘0)1) = ∅
117	114, 116	eqtrdi 2814	. . . . 5 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) = ∅)
118		0ss 4352	. . . . 5 ⊢ ∅ ⊆ (◡𝑓 “ {(𝑓‘1)})
119	117, 118	eqsstrdi 3956	. . . 4 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
120	119	ralrimivw 3161	. . 3 ⊢ (𝐾 = 0 → ∀𝑓 ∈ (𝑅 ↑m (1...1))(1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
121	120, 51	syl5 35	. 2 ⊢ (𝜑 → (𝐾 = 0 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓))
122		elnn0 12618	. . 3 ⊢ (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
123	41, 122	sylib 221	. 2 ⊢ (𝜑 → (𝐾 ∈ ℕ ∨ 𝐾 = 0))
124	112, 121, 123	mpjaod 874	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2146  ∀wral 3079  ∃wrex 3089  Vcvv 3453   ⊆ wss 3899  ∅c0 4280  {csn 4587   class class class wbr 5105  ^◡ccnv 5653   “ cima 5657   Fn wfn 6541  ⟶wf 6542  ‘cfv 6546  (class class class)co 7427   ↑_m cmap 8858  Fincfn 8984  0cc0 11210  1c1 11211   + caddc 11213  ℕcn 12345  2c2 12407  ℕ₀cn0 12616  ℤcz 12703  ℤ_≥cuz 12975  ...cfz 13649  ♯chash 14484  APcvdwa 17155   MonoAP cvdwm 17156

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-oadd 8491  df-er 8728  df-map 8860  df-pm 8861  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9992  df-card 10030  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-nn 12346  df-2 12415  df-n0 12617  df-xnn0 12690  df-z 12704  df-uz 12976  df-rp 13131  df-fz 13650  df-hash 14485  df-vdwap 17158  df-vdwmc 17159  df-vdwpc 17160

This theorem is used by:  vdw  17184