Theorem vdwlem13 16905

Description: Lemma for vdw 16906. 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 12823	. . 3 ⊢ (𝐾 ∈ ℕ ↔ (𝐾 = 1 ∨ 𝐾 ∈ (ℤ≥‘2)))
2		vdw.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Fin)
3		ovex 7379	. . . . . . . . . 10 ⊢ (1...1) ∈ V
4		elmapg 8763	. . . . . . . . . 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 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑m (1...1))) → 𝑓:(1...1)⟶𝑅)
7		1nn 12136	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
8		vdwap1 16889	. . . . . . . . . 10 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘1)1) = {1})
9	7, 7, 8	mp2an 692	. . . . . . . . 9 ⊢ (1(AP‘1)1) = {1}
10		1z 12502	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
11		elfz3 13434	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ (1...1))
12	10, 11	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 1 ∈ (1...1))
13		eqidd 2732	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (𝑓‘1) = (𝑓‘1))
14		ffn 6651	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...1)⟶𝑅 → 𝑓 Fn (1...1))
15	14	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 𝑓 Fn (1...1))
16		fniniseg 6993	. . . . . . . . . . . 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 713	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 1 ∈ (◡𝑓 “ {(𝑓‘1)}))
19	18	snssd 4758	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → {1} ⊆ (◡𝑓 “ {(𝑓‘1)}))
20	9, 19	eqsstrid 3968	. . . . . . . 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 3124	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ (𝑅 ↑m (1...1))(1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
23		fveq2 6822	. . . . . . . . 9 ⊢ (𝐾 = 1 → (AP‘𝐾) = (AP‘1))
24	23	oveqd 7363	. . . . . . . 8 ⊢ (𝐾 = 1 → (1(AP‘𝐾)1) = (1(AP‘1)1))
25	24	sseq1d 3961	. . . . . . 7 ⊢ (𝐾 = 1 → ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
26	25	ralbidv 3155	. . . . . 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 247	. . . . 5 ⊢ (𝜑 → (𝐾 = 1 → ∀𝑓 ∈ (𝑅 ↑m (1...1))(1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
28		oveq1 7353	. . . . . . . . . . . 12 ⊢ (𝑎 = 1 → (𝑎(AP‘𝐾)𝑑) = (1(AP‘𝐾)𝑑))
29	28	sseq1d 3961	. . . . . . . . . . 11 ⊢ (𝑎 = 1 → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)})))
30		oveq2 7354	. . . . . . . . . . . 12 ⊢ (𝑑 = 1 → (1(AP‘𝐾)𝑑) = (1(AP‘𝐾)1))
31	30	sseq1d 3961	. . . . . . . . . . 11 ⊢ (𝑑 = 1 → ((1(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
32	29, 31	rspc2ev 3585	. . . . . . . . . 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 1453	. . . . . . . . 9 ⊢ ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}))
34		fvex 6835	. . . . . . . . . 10 ⊢ (𝑓‘1) ∈ V
35		sneq 4583	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑓‘1) → {𝑐} = {(𝑓‘1)})
36	35	imaeq2d 6008	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑓‘1) → (◡𝑓 “ {𝑐}) = (◡𝑓 “ {(𝑓‘1)}))
37	36	sseq2d 3962	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑓‘1) → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)})))
38	37	2rexbidv 3197	. . . . . . . . . 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 3556	. . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑m (1...1))) → 𝐾 ∈ ℕ0)
43	3, 42, 6	vdwmc 16890	. . . . . . . 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 246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑m (1...1))) → ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → 𝐾 MonoAP 𝑓))
45	44	ralimdva 3144	. . . . . 6 ⊢ (𝜑 → (∀𝑓 ∈ (𝑅 ↑m (1...1))(1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → ∀𝑓 ∈ (𝑅 ↑m (1...1))𝐾 MonoAP 𝑓))
46		oveq2 7354	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
47	46	oveq2d 7362	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑅 ↑m (1...𝑛)) = (𝑅 ↑m (1...1)))
48	47	raleqdv 3292	. . . . . . . 8 ⊢ (𝑛 = 1 → (∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑m (1...1))𝐾 MonoAP 𝑓))
49	48	rspcev 3572	. . . . . . 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 690	. . . . . 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 5092	. . . . . . . 8 ⊢ (𝑥 = 2 → (𝑥 MonoAP 𝑓 ↔ 2 MonoAP 𝑓))
54	53	rexralbidv 3198	. . . . . . 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 3155	. . . . . 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 5092	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝑥 MonoAP 𝑓 ↔ 𝑘 MonoAP 𝑓))
57	56	rexralbidv 3198	. . . . . . 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 3155	. . . . . 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 5092	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 MonoAP 𝑓 ↔ (𝑘 + 1) MonoAP 𝑓))
60	59	rexralbidv 3198	. . . . . . 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 3155	. . . . . 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 5092	. . . . . . . 8 ⊢ (𝑥 = 𝐾 → (𝑥 MonoAP 𝑓 ↔ 𝐾 MonoAP 𝑓))
63	62	rexralbidv 3198	. . . . . . 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 3155	. . . . . 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 14263	. . . . . . . . 9 ⊢ (𝑟 ∈ Fin → (♯‘𝑟) ∈ ℕ0)
66		nn0p1nn 12420	. . . . . . . . 9 ⊢ ((♯‘𝑟) ∈ ℕ0 → ((♯‘𝑟) + 1) ∈ ℕ)
67	65, 66	syl 17	. . . . . . . 8 ⊢ (𝑟 ∈ Fin → ((♯‘𝑟) + 1) ∈ ℕ)
68		simpll 766	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑟 ∈ Fin)
69		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1))))
70		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑟 ∈ V
71		ovex 7379	. . . . . . . . . . . . 13 ⊢ (1...((♯‘𝑟) + 1)) ∈ V
72	70, 71	elmap 8795	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1))) ↔ 𝑓:(1...((♯‘𝑟) + 1))⟶𝑟)
73	69, 72	sylib 218	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓:(1...((♯‘𝑟) + 1))⟶𝑟)
74		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → ¬ 2 MonoAP 𝑓)
75	68, 73, 74	vdwlem12 16904	. . . . . . . . . 10 ⊢ ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓)
76		iman 401	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))) → 2 MonoAP 𝑓)
78	77	ralrimiva 3124	. . . . . . . 8 ⊢ (𝑟 ∈ Fin → ∀𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓)
79		oveq2 7354	. . . . . . . . . . 11 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (1...𝑛) = (1...((♯‘𝑟) + 1)))
80	79	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (𝑟 ↑m (1...𝑛)) = (𝑟 ↑m (1...((♯‘𝑟) + 1))))
81	80	raleqdv 3292	. . . . . . . . 9 ⊢ (𝑛 = ((♯‘𝑟) + 1) → (∀𝑓 ∈ (𝑟 ↑m (1...𝑛))2 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑟 ↑m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓))
82	81	rspcev 3572	. . . . . . . 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 3049	. . . . . 6 ⊢ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑m (1...𝑛))2 MonoAP 𝑓
85		oveq1 7353	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → (𝑟 ↑m (1...𝑛)) = (𝑠 ↑m (1...𝑛)))
86	85	raleqdv 3292	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (∀𝑓 ∈ (𝑟 ↑m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑m (1...𝑛))𝑘 MonoAP 𝑓))
87	86	rexbidv 3156	. . . . . . . . 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 7354	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
89	88	oveq2d 7362	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝑠 ↑m (1...𝑛)) = (𝑠 ↑m (1...𝑚)))
90	89	raleqdv 3292	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠 ↑m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑓))
91		breq2 5093	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑘 MonoAP 𝑓 ↔ 𝑘 MonoAP 𝑔))
92	91	cbvralvw 3210	. . . . . . . . . . 11 ⊢ (∀𝑓 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑔)
93	90, 92	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠 ↑m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑔))
94	93	cbvrexvw 3211	. . . . . . . . 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 287	. . . . . . . 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 3210	. . . . . . 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 768	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑔) → 𝑟 ∈ Fin)
98		simpll 766	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑m (1...𝑚))𝑘 MonoAP 𝑔) → 𝑘 ∈ (ℤ≥‘2))
99		simpr 484	. . . . . . . . . . 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 3078	. . . . . . . . . . 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 234	. . . . . . . . . 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 16903	. . . . . . . . 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 412	. . . . . . . 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 3132	. . . . . . 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 242	. . . . . 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 12808	. . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘2) → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑m (1...𝑛))𝐾 MonoAP 𝑓)
107		oveq1 7353	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 ↑m (1...𝑛)) = (𝑅 ↑m (1...𝑛)))
108	107	raleqdv 3292	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∀𝑓 ∈ (𝑟 ↑m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓))
109	108	rexbidv 3156	. . . . . 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 3568	. . . . 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 859	. . 3 ⊢ (𝜑 → ((𝐾 = 1 ∨ 𝐾 ∈ (ℤ≥‘2)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓))
113	1, 112	biimtrid 242	. 2 ⊢ (𝜑 → (𝐾 ∈ ℕ → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓))
114		fveq2 6822	. . . . . . 7 ⊢ (𝐾 = 0 → (AP‘𝐾) = (AP‘0))
115	114	oveqd 7363	. . . . . 6 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) = (1(AP‘0)1))
116		vdwap0 16888	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘0)1) = ∅)
117	7, 7, 116	mp2an 692	. . . . . 6 ⊢ (1(AP‘0)1) = ∅
118	115, 117	eqtrdi 2782	. . . . 5 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) = ∅)
119		0ss 4347	. . . . 5 ⊢ ∅ ⊆ (◡𝑓 “ {(𝑓‘1)})
120	118, 119	eqsstrdi 3974	. . . 4 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
121	120	ralrimivw 3128	. . 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 12383	. . 3 ⊢ (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
124	41, 123	sylib 218	. 2 ⊢ (𝜑 → (𝐾 ∈ ℕ ∨ 𝐾 = 0))
125	113, 122, 124	mpjaod 860	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  ∅c0 4280  {csn 4573   class class class wbr 5089  ^◡ccnv 5613   “ cima 5617   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750  Fincfn 8869  0cc0 11006  1c1 11007   + caddc 11009  ℕcn 12125  2c2 12180  ℕ₀cn0 12381  ℤcz 12468  ℤ_≥cuz 12732  ...cfz 13407  ♯chash 14237  APcvdwa 16877   MonoAP cvdwm 16878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-n0 12382  df-xnn0 12455  df-z 12469  df-uz 12733  df-rp 12891  df-fz 13408  df-hash 14238  df-vdwap 16880  df-vdwmc 16881  df-vdwpc 16882

This theorem is referenced by:  vdw  16906