Theorem nninfsellemeqinf 15506

Description: Lemma for nninfsel 15507. (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 15503	. . . . . 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 5694	. . . 4 ⊢ (𝜑 → (𝐸‘𝑄) ∈ ℕ∞)
6		nninff 7181	. . . 4 ⊢ ((𝐸‘𝑄) ∈ ℕ∞ → (𝐸‘𝑄):ω⟶2o)
7	5, 6	syl 14	. . 3 ⊢ (𝜑 → (𝐸‘𝑄):ω⟶2o)
8	7	ffnd 5404	. 2 ⊢ (𝜑 → (𝐸‘𝑄) Fn ω)
9		1onn 6573	. . . . 5 ⊢ 1o ∈ ω
10		fnconstg 5451	. . . . 5 ⊢ (1o ∈ ω → (ω × {1o}) Fn ω)
11	9, 10	ax-mp 5	. . . 4 ⊢ (ω × {1o}) Fn ω
12		fconstmpt 4706	. . . . 5 ⊢ (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
13	12	fneq1i 5348	. . . 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 3578	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → if(𝑖 ∈ 𝑗, 1o, ∅) = if(𝑖 ∈ 𝑘, 1o, ∅))
18	17	mpteq2dv 4120	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅)))
19	18	fveq2d 5558	. . . . . . 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 15505	. . . . . . . 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 4627	. . . . . . . 8 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
30	29	ad2antlr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → suc 𝑗 ∈ ω)
31		elnn 4638	. . . . . . 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 2869	. . . . 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 3564	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) = 1o)
36		omex 4625	. . . . . . 7 ⊢ ω ∈ V
37	36	mptex 5784	. . . . . 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 5553	. . . . . . . . . 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 3578	. . . . . . 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 4120	. . . . . 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 5633	. . . . 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 4433	. . . . . . 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 3578	. . . 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 5638	. . 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 5633	. . . . 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 5658	1 ⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ 1o))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∀wral 2472  Vcvv 2760  ∅c0 3446  ifcif 3557  {csn 3618   ↦ cmpt 4090  suc csuc 4396  ωcom 4622   × cxp 4657   Fn wfn 5249  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918  1_oc1o 6462  2_oc2o 6463   ↑_𝑚 cmap 6702  ℕ_∞xnninf 7178

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 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1o 6469  df-2o 6470  df-map 6704  df-nninf 7179

This theorem is referenced by:  nninfsel  15507