Theorem nninfsellemeq 16616

Description: Lemma for nninfsel 16619. (Contributed by Jim Kingdon, 9-Aug-2022.)

Hypotheses

Ref	Expression
nninfsel.e	⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))
nninfsel.q	⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))
nninfsel.1	⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o)
nninfsel.n	⊢ (𝜑 → 𝑁 ∈ ω)
nninfsel.qk	⊢ (𝜑 → ∀𝑘 ∈ 𝑁 (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
nninfsel.qn	⊢ (𝜑 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) = ∅)

Assertion

Ref	Expression
nninfsellemeq	⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)))

Distinct variable groups:   𝑖,𝑁,𝑘,𝑛   𝑄,𝑛,𝑘,𝑞   𝜑,𝑖,𝑛   𝑖,𝑞

Allowed substitution hints:   𝜑(𝑘,𝑞)   𝑄(𝑖)   𝐸(𝑖,𝑘,𝑛,𝑞)   𝑁(𝑞)

Proof of Theorem nninfsellemeq

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nninfsel.e	. . . . 5 ⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))
2	1	nninfself 16615	. . . 4 ⊢ 𝐸:(2o ↑𝑚 ℕ∞)⟶ℕ∞
3	2	a1i 9	. . 3 ⊢ (𝜑 → 𝐸:(2o ↑𝑚 ℕ∞)⟶ℕ∞)
4		nninfsel.q	. . 3 ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))
5	3, 4	ffvelcdmd 5783	. 2 ⊢ (𝜑 → (𝐸‘𝑄) ∈ ℕ∞)
6		nninfsel.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ω)
7		fveq1 5638	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑄 → (𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))))
8	7	eqeq1d 2240	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → ((𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o))
9	8	ralbidv 2532	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o ↔ ∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o))
10	9	ifbid 3627	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) = if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))
11	10	mpteq2dv 4180	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))
12		omex 4691	. . . . . . . 8 ⊢ ω ∈ V
13	12	mptex 5879	. . . . . . 7 ⊢ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)) ∈ V
14	11, 1, 13	fvmpt 5723	. . . . . 6 ⊢ (𝑄 ∈ (2o ↑𝑚 ℕ∞) → (𝐸‘𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))
15	4, 14	syl 14	. . . . 5 ⊢ (𝜑 → (𝐸‘𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))
16	15	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → (𝐸‘𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))
17		simpr 110	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 = 𝑗) → 𝑛 = 𝑗)
18		simplr 529	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 = 𝑗) → 𝑗 ∈ 𝑁)
19	17, 18	eqeltrd 2308	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 = 𝑗) → 𝑛 ∈ 𝑁)
20		nnord 4710	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → Ord 𝑁)
21		vex 2805	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
22		ordelsuc 4603	. . . . . . . . . 10 ⊢ ((𝑛 ∈ V ∧ Ord 𝑁) → (𝑛 ∈ 𝑁 ↔ suc 𝑛 ⊆ 𝑁))
23	21, 22	mpan 424	. . . . . . . . 9 ⊢ (Ord 𝑁 → (𝑛 ∈ 𝑁 ↔ suc 𝑛 ⊆ 𝑁))
24	6, 20, 23	3syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ 𝑁 ↔ suc 𝑛 ⊆ 𝑁))
25	24	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 = 𝑗) → (𝑛 ∈ 𝑁 ↔ suc 𝑛 ⊆ 𝑁))
26	19, 25	mpbid 147	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 = 𝑗) → suc 𝑛 ⊆ 𝑁)
27		nninfsel.qk	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝑁 (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
28	27	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 = 𝑗) → ∀𝑘 ∈ 𝑁 (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
29		ssralv 3291	. . . . . 6 ⊢ (suc 𝑛 ⊆ 𝑁 → (∀𝑘 ∈ 𝑁 (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o → ∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o))
30	26, 28, 29	sylc 62	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 = 𝑗) → ∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
31	30	iftrued 3612	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑁) ∧ 𝑛 = 𝑗) → if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) = 1o)
32		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
33	6	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑁 ∈ ω)
34		elnn 4704	. . . . 5 ⊢ ((𝑗 ∈ 𝑁 ∧ 𝑁 ∈ ω) → 𝑗 ∈ ω)
35	32, 33, 34	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ ω)
36		1onn 6687	. . . . 5 ⊢ 1o ∈ ω
37	36	a1i 9	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 1o ∈ ω)
38	16, 31, 35, 37	fvmptd 5727	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → ((𝐸‘𝑄)‘𝑗) = 1o)
39	38	ralrimiva 2605	. 2 ⊢ (𝜑 → ∀𝑗 ∈ 𝑁 ((𝐸‘𝑄)‘𝑗) = 1o)
40	21	sucid 4514	. . . . . . 7 ⊢ 𝑛 ∈ suc 𝑛
41	40	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → 𝑛 ∈ suc 𝑛)
42		1n0 6599	. . . . . . . 8 ⊢ 1o ≠ ∅
43	42	nesymi 2448	. . . . . . 7 ⊢ ¬ ∅ = 1o
44		simpr 110	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
45	44	eleq2d 2301	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ 𝑁))
46	45	ifbid 3627	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ 𝑁, 1o, ∅))
47	46	mpteq2dv 4180	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)))
48	47	fveq2d 5643	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))))
49		nninfsel.qn	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) = ∅)
50	49	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) = ∅)
51	48, 50	eqtrd 2264	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = ∅)
52	51	eqeq1d 2240	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o ↔ ∅ = 1o))
53	43, 52	mtbiri 681	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → ¬ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o)
54		elequ2 2207	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝑖 ∈ 𝑘 ↔ 𝑖 ∈ 𝑛))
55	54	ifbid 3627	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → if(𝑖 ∈ 𝑘, 1o, ∅) = if(𝑖 ∈ 𝑛, 1o, ∅))
56	55	mpteq2dv 4180	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
57	56	fveq2d 5643	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))
58	57	eqeq1d 2240	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o))
59	58	notbid 673	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (¬ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o ↔ ¬ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o))
60	59	rspcev 2910	. . . . . 6 ⊢ ((𝑛 ∈ suc 𝑛 ∧ ¬ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o) → ∃𝑘 ∈ suc 𝑛 ¬ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
61	41, 53, 60	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → ∃𝑘 ∈ suc 𝑛 ¬ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
62		rexnalim 2521	. . . . 5 ⊢ (∃𝑘 ∈ suc 𝑛 ¬ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o → ¬ ∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
63	61, 62	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → ¬ ∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
64	63	iffalsed 3615	. . 3 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) = ∅)
65		peano1 4692	. . . 4 ⊢ ∅ ∈ ω
66	65	a1i 9	. . 3 ⊢ (𝜑 → ∅ ∈ ω)
67	15, 64, 6, 66	fvmptd 5727	. 2 ⊢ (𝜑 → ((𝐸‘𝑄)‘𝑁) = ∅)
68	5, 6, 39, 67	nnnninfeq 7326	1 ⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  Vcvv 2802   ⊆ wss 3200  ∅c0 3494  ifcif 3605   ↦ cmpt 4150  Ord word 4459  suc csuc 4462  ωcom 4688  ⟶wf 5322  ‘cfv 5326  (class class class)co 6017  1_oc1o 6574  2_oc2o 6575   ↑_𝑚 cmap 6816  ℕ_∞xnninf 7317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1o 6581  df-2o 6582  df-map 6818  df-nninf 7318

This theorem is referenced by:  nninfsellemqall  16617