Theorem nninfsellemeqinf 15576

Description: Lemma for nninfsel 15577. (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)

Assertion

Ref	Expression
nninfsellemeqinf	⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ 1o))

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

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

Proof of Theorem nninfsellemeqinf

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nninfsel.e	. . . . . . 7 ⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))
2	1	nninfself 15573	. . . . . 6 ⊢ 𝐸:(2o ↑𝑚 ℕ∞)⟶ℕ∞
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → 𝐸:(2o ↑𝑚 ℕ∞)⟶ℕ∞)
4		nninfsel.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞))
5	3, 4	ffvelcdmd 5695	. . . 4 ⊢ (𝜑 → (𝐸‘𝑄) ∈ ℕ∞)
6		nninff 7183	. . . 4 ⊢ ((𝐸‘𝑄) ∈ ℕ∞ → (𝐸‘𝑄):ω⟶2o)
7	5, 6	syl 14	. . 3 ⊢ (𝜑 → (𝐸‘𝑄):ω⟶2o)
8	7	ffnd 5405	. 2 ⊢ (𝜑 → (𝐸‘𝑄) Fn ω)
9		1onn 6575	. . . . 5 ⊢ 1o ∈ ω
10		fnconstg 5452	. . . . 5 ⊢ (1o ∈ ω → (ω × {1o}) Fn ω)
11	9, 10	ax-mp 5	. . . 4 ⊢ (ω × {1o}) Fn ω
12		fconstmpt 4707	. . . . 5 ⊢ (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
13	12	fneq1i 5349	. . . 4 ⊢ ((ω × {1o}) Fn ω ↔ (𝑖 ∈ ω ↦ 1o) Fn ω)
14	11, 13	mpbi 145	. . 3 ⊢ (𝑖 ∈ ω ↦ 1o) Fn ω
15	14	a1i 9	. 2 ⊢ (𝜑 → (𝑖 ∈ ω ↦ 1o) Fn ω)
16		elequ2 2169	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑖 ∈ 𝑗 ↔ 𝑖 ∈ 𝑘))
17	16	ifbid 3579	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → if(𝑖 ∈ 𝑗, 1o, ∅) = if(𝑖 ∈ 𝑘, 1o, ∅))
18	17	mpteq2dv 4121	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅)))
19	18	fveq2d 5559	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))))
20	19	eqeq1d 2202	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o))
21	4	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → 𝑄 ∈ (2o ↑𝑚 ℕ∞))
22		nninfsel.1	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o)
23	22	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝑄‘(𝐸‘𝑄)) = 1o)
24		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
25	1, 21, 23, 24	nninfsellemqall 15575	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅))) = 1o)
26	25	ralrimiva 2567	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅))) = 1o)
27	26	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅))) = 1o)
28		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ suc 𝑗)
29		peano2 4628	. . . . . . . 8 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
30	29	ad2antlr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → suc 𝑗 ∈ ω)
31		elnn 4639	. . . . . . 7 ⊢ ((𝑘 ∈ suc 𝑗 ∧ suc 𝑗 ∈ ω) → 𝑘 ∈ ω)
32	28, 30, 31	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ ω)
33	20, 27, 32	rspcdva 2870	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
34	33	ralrimiva 2567	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
35	34	iftrued 3565	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) = 1o)
36		omex 4626	. . . . . . 7 ⊢ ω ∈ V
37	36	mptex 5785	. . . . . 6 ⊢ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)) ∈ V
38	37	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)) ∈ V)
39		fveq1 5554	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → (𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))))
40	39	eqeq1d 2202	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → ((𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o))
41	40	ralbidv 2494	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o ↔ ∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o))
42	41	ifbid 3579	. . . . . . 7 ⊢ (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, ∅))
43	42	mpteq2dv 4121	. . . . . 6 ⊢ (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, ∅)))
44	43, 1	fvmptg 5634	. . . . 5 ⊢ ((𝑄 ∈ (2o ↑𝑚 ℕ∞) ∧ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)) ∈ V) → (𝐸‘𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))
45	21, 38, 44	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝐸‘𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))
46		suceq 4434	. . . . . . 7 ⊢ (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
47	46	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → suc 𝑛 = suc 𝑗)
48	47	raleqdv 2696	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → (∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o ↔ ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o))
49	48	ifbid 3579	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) = if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))
50	35, 9	eqeltrdi 2284	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ ω)
51	45, 49, 24, 50	fvmptd 5639	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ((𝐸‘𝑄)‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))
52		eqidd 2194	. . . . . 6 ⊢ (𝑖 = 𝑗 → 1o = 1o)
53		eqid 2193	. . . . . 6 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
54	52, 53	fvmptg 5634	. . . . 5 ⊢ ((𝑗 ∈ ω ∧ 1o ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
55	9, 54	mpan2 425	. . . 4 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
56	55	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
57	35, 51, 56	3eqtr4d 2236	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ((𝐸‘𝑄)‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
58	8, 15, 57	eqfnfvd 5659	1 ⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ 1o))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∀wral 2472  Vcvv 2760  ∅c0 3447  ifcif 3558  {csn 3619   ↦ cmpt 4091  suc csuc 4397  ωcom 4623   × cxp 4658   Fn wfn 5250  ⟶wf 5251  ‘cfv 5255  (class class class)co 5919  1_oc1o 6464  2_oc2o 6465   ↑_𝑚 cmap 6704  ℕ_∞xnninf 7180

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1o 6471  df-2o 6472  df-map 6706  df-nninf 7181

This theorem is referenced by:  nninfsel  15577