Theorem nninfsellemeq 16723

Description: Lemma for nninfsel 16726. (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 16722	. . . 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 5791	. 2 ⊢ (𝜑 → (𝐸‘𝑄) ∈ ℕ∞)
6		nninfsel.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ω)
7		fveq1 5647	. . . . . . . . . . 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 2533	. . . . . . . . 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 3631	. . . . . . . 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 4185	. . . . . . 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 4697	. . . . . . . 8 ⊢ ω ∈ V
13	12	mptex 5890	. . . . . . 7 ⊢ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)) ∈ V
14	11, 1, 13	fvmpt 5732	. . . . . 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 4716	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → Ord 𝑁)
21		vex 2806	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
22		ordelsuc 4609	. . . . . . . . . 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 3292	. . . . . 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 3616	. . . 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 4710	. . . . 5 ⊢ ((𝑗 ∈ 𝑁 ∧ 𝑁 ∈ ω) → 𝑗 ∈ ω)
35	32, 33, 34	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ ω)
36		1onn 6731	. . . . 5 ⊢ 1o ∈ ω
37	36	a1i 9	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 1o ∈ ω)
38	16, 31, 35, 37	fvmptd 5736	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → ((𝐸‘𝑄)‘𝑗) = 1o)
39	38	ralrimiva 2606	. 2 ⊢ (𝜑 → ∀𝑗 ∈ 𝑁 ((𝐸‘𝑄)‘𝑗) = 1o)
40	21	sucid 4520	. . . . . . 7 ⊢ 𝑛 ∈ suc 𝑛
41	40	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → 𝑛 ∈ suc 𝑛)
42		1n0 6643	. . . . . . . 8 ⊢ 1o ≠ ∅
43	42	nesymi 2449	. . . . . . 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 3631	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ 𝑁, 1o, ∅))
47	46	mpteq2dv 4185	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)))
48	47	fveq2d 5652	. . . . . . . . 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 682	. . . . . 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 3631	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → if(𝑖 ∈ 𝑘, 1o, ∅) = if(𝑖 ∈ 𝑛, 1o, ∅))
56	55	mpteq2dv 4185	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
57	56	fveq2d 5652	. . . . . . . . 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 2911	. . . . . 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 2522	. . . . 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 3619	. . 3 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) = ∅)
65		peano1 4698	. . . 4 ⊢ ∅ ∈ ω
66	65	a1i 9	. . 3 ⊢ (𝜑 → ∅ ∈ ω)
67	15, 64, 6, 66	fvmptd 5736	. 2 ⊢ (𝜑 → ((𝐸‘𝑄)‘𝑁) = ∅)
68	5, 6, 39, 67	nnnninfeq 7370	1 ⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  Vcvv 2803   ⊆ wss 3201  ∅c0 3496  ifcif 3607   ↦ cmpt 4155  Ord word 4465  suc csuc 4468  ωcom 4694  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  1_oc1o 6618  2_oc2o 6619   ↑_𝑚 cmap 6860  ℕ_∞xnninf 7361

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1o 6625  df-2o 6626  df-map 6862  df-nninf 7362

This theorem is referenced by:  nninfsellemqall  16724