Theorem nninfsellemeqinf 12796

Description: Lemma for nninfsel 12797. (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 12793	. . . . . 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	ffvelrnd 5488	. . . 4 ⊢ (𝜑 → (𝐸‘𝑄) ∈ ℕ∞)
6		nninff 12782	. . . 4 ⊢ ((𝐸‘𝑄) ∈ ℕ∞ → (𝐸‘𝑄):ω⟶2o)
7	5, 6	syl 14	. . 3 ⊢ (𝜑 → (𝐸‘𝑄):ω⟶2o)
8	7	ffnd 5209	. 2 ⊢ (𝜑 → (𝐸‘𝑄) Fn ω)
9		1onn 6346	. . . . 5 ⊢ 1o ∈ ω
10		fnconstg 5256	. . . . 5 ⊢ (1o ∈ ω → (ω × {1o}) Fn ω)
11	9, 10	ax-mp 7	. . . 4 ⊢ (ω × {1o}) Fn ω
12		fconstmpt 4524	. . . . 5 ⊢ (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
13	12	fneq1i 5153	. . . 4 ⊢ ((ω × {1o}) Fn ω ↔ (𝑖 ∈ ω ↦ 1o) Fn ω)
14	11, 13	mpbi 144	. . 3 ⊢ (𝑖 ∈ ω ↦ 1o) Fn ω
15	14	a1i 9	. 2 ⊢ (𝜑 → (𝑖 ∈ ω ↦ 1o) Fn ω)
16		elequ2 1659	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑖 ∈ 𝑗 ↔ 𝑖 ∈ 𝑘))
17	16	ifbid 3440	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → if(𝑖 ∈ 𝑗, 1o, ∅) = if(𝑖 ∈ 𝑘, 1o, ∅))
18	17	mpteq2dv 3959	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅)))
19	18	fveq2d 5357	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))))
20	19	eqeq1d 2108	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o))
21	4	adantr 272	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → 𝑄 ∈ (2o ↑𝑚 ℕ∞))
22		nninfsel.1	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o)
23	22	adantr 272	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝑄‘(𝐸‘𝑄)) = 1o)
24		simpr 109	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
25	1, 21, 23, 24	nninfsellemqall 12795	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅))) = 1o)
26	25	ralrimiva 2464	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅))) = 1o)
27	26	ad2antrr 475	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1o, ∅))) = 1o)
28		simpr 109	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ suc 𝑗)
29		peano2 4447	. . . . . . . 8 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
30	29	ad2antlr 476	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → suc 𝑗 ∈ ω)
31		elnn 4457	. . . . . . 7 ⊢ ((𝑘 ∈ suc 𝑗 ∧ suc 𝑗 ∈ ω) → 𝑘 ∈ ω)
32	28, 30, 31	syl2anc 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ ω)
33	20, 27, 32	rspcdva 2749	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
34	33	ralrimiva 2464	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o)
35	34	iftrued 3428	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) = 1o)
36		omex 4445	. . . . . . 7 ⊢ ω ∈ V
37	36	mptex 5578	. . . . . 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 5352	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → (𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))))
40	39	eqeq1d 2108	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → ((𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o))
41	40	ralbidv 2396	. . . . . . . 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 3440	. . . . . . 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 3959	. . . . . 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 5429	. . . . 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 406	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝐸‘𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)))
46		suceq 4262	. . . . . . 7 ⊢ (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
47	46	adantl 273	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → suc 𝑛 = suc 𝑗)
48	47	raleqdv 2590	. . . . 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 3440	. . . 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	syl6eqel 2190	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ ω)
51	45, 49, 24, 50	fvmptd 5434	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ((𝐸‘𝑄)‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))
52		eqidd 2101	. . . . . 6 ⊢ (𝑖 = 𝑗 → 1o = 1o)
53		eqid 2100	. . . . . 6 ⊢ (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
54	52, 53	fvmptg 5429	. . . . 5 ⊢ ((𝑗 ∈ ω ∧ 1o ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
55	9, 54	mpan2 419	. . . 4 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
56	55	adantl 273	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
57	35, 51, 56	3eqtr4d 2142	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ((𝐸‘𝑄)‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
58	8, 15, 57	eqfnfvd 5453	1 ⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ 1o))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1299   ∈ wcel 1448  ∀wral 2375  Vcvv 2641  ∅c0 3310  ifcif 3421  {csn 3474   ↦ cmpt 3929  suc csuc 4225  ωcom 4442   × cxp 4475   Fn wfn 5054  ⟶wf 5055  ‘cfv 5059  (class class class)co 5706  1_oc1o 6236  2_oc2o 6237   ↑_𝑚 cmap 6472  ℕ_∞xnninf 6917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1o 6243  df-2o 6244  df-map 6474  df-nninf 6919

This theorem is referenced by:  nninfsel  12797