Theorem nninfsellemeqinf 11552

Description: Lemma for nninfsel 11553. (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 11549	. . . . . 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 5419	. . . 4 ⊢ (𝜑 → (𝐸‘𝑄) ∈ ℕ∞)
6		nninff 11538	. . . 4 ⊢ ((𝐸‘𝑄) ∈ ℕ∞ → (𝐸‘𝑄):ω⟶2𝑜)
7	5, 6	syl 14	. . 3 ⊢ (𝜑 → (𝐸‘𝑄):ω⟶2𝑜)
8	7	ffnd 5148	. 2 ⊢ (𝜑 → (𝐸‘𝑄) Fn ω)
9		1onn 6259	. . . . 5 ⊢ 1𝑜 ∈ ω
10		fnconstg 5192	. . . . 5 ⊢ (1𝑜 ∈ ω → (ω × {1𝑜}) Fn ω)
11	9, 10	ax-mp 7	. . . 4 ⊢ (ω × {1𝑜}) Fn ω
12		fconstmpt 4473	. . . . 5 ⊢ (ω × {1𝑜}) = (𝑖 ∈ ω ↦ 1𝑜)
13	12	fneq1i 5094	. . . 4 ⊢ ((ω × {1𝑜}) Fn ω ↔ (𝑖 ∈ ω ↦ 1𝑜) Fn ω)
14	11, 13	mpbi 143	. . 3 ⊢ (𝑖 ∈ ω ↦ 1𝑜) Fn ω
15	14	a1i 9	. 2 ⊢ (𝜑 → (𝑖 ∈ ω ↦ 1𝑜) Fn ω)
16		elequ2 1648	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑖 ∈ 𝑗 ↔ 𝑖 ∈ 𝑘))
17	16	ifbid 3408	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → if(𝑖 ∈ 𝑗, 1𝑜, ∅) = if(𝑖 ∈ 𝑘, 1𝑜, ∅))
18	17	mpteq2dv 3921	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1𝑜, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅)))
19	18	fveq2d 5293	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1𝑜, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))))
20	19	eqeq1d 2096	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1𝑜, ∅))) = 1𝑜 ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
21	4	adantr 270	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞))
22		nninfsel.1	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1𝑜)
23	22	adantr 270	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝑄‘(𝐸‘𝑄)) = 1𝑜)
24		simpr 108	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
25	1, 21, 23, 24	nninfsellemqall 11551	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1𝑜, ∅))) = 1𝑜)
26	25	ralrimiva 2446	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1𝑜, ∅))) = 1𝑜)
27	26	ad2antrr 472	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑗, 1𝑜, ∅))) = 1𝑜)
28		simpr 108	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ suc 𝑗)
29		peano2 4400	. . . . . . . 8 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
30	29	ad2antlr 473	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → suc 𝑗 ∈ ω)
31		elnn 4410	. . . . . . 7 ⊢ ((𝑘 ∈ suc 𝑗 ∧ suc 𝑗 ∈ ω) → 𝑘 ∈ ω)
32	28, 30, 31	syl2anc 403	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ ω)
33	20, 27, 32	rspcdva 2727	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)
34	33	ralrimiva 2446	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)
35	34	iftrued 3396	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) = 1𝑜)
36		omex 4398	. . . . . . 7 ⊢ ω ∈ V
37	36	mptex 5505	. . . . . 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 5288	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → (𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))))
40	39	eqeq1d 2096	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → ((𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
41	40	ralbidv 2380	. . . . . . . 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 3408	. . . . . . 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 3921	. . . . . 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 5364	. . . . 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 403	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → (𝐸‘𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)))
46		suceq 4220	. . . . . . 7 ⊢ (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
47	46	adantl 271	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → suc 𝑛 = suc 𝑗)
48	47	raleqdv 2568	. . . . 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 3408	. . . 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 2178	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ ω)
51	45, 49, 24, 50	fvmptd 5369	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ((𝐸‘𝑄)‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
52		eqidd 2089	. . . . . 6 ⊢ (𝑖 = 𝑗 → 1𝑜 = 1𝑜)
53		eqid 2088	. . . . . 6 ⊢ (𝑖 ∈ ω ↦ 1𝑜) = (𝑖 ∈ ω ↦ 1𝑜)
54	52, 53	fvmptg 5364	. . . . 5 ⊢ ((𝑗 ∈ ω ∧ 1𝑜 ∈ ω) → ((𝑖 ∈ ω ↦ 1𝑜)‘𝑗) = 1𝑜)
55	9, 54	mpan2 416	. . . 4 ⊢ (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1𝑜)‘𝑗) = 1𝑜)
56	55	adantl 271	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ 1𝑜)‘𝑗) = 1𝑜)
57	35, 51, 56	3eqtr4d 2130	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ ω) → ((𝐸‘𝑄)‘𝑗) = ((𝑖 ∈ ω ↦ 1𝑜)‘𝑗))
58	8, 15, 57	eqfnfvd 5384	1 ⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ 1𝑜))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1289   ∈ wcel 1438  ∀wral 2359  Vcvv 2619  ∅c0 3284  ifcif 3389  {csn 3441   ↦ cmpt 3891  suc csuc 4183  ωcom 4395   × cxp 4426   Fn wfn 4997  ⟶wf 4998  ‘cfv 5002  (class class class)co 5634  1_𝑜c1o 6156  2_𝑜c2o 6157   ↑_𝑚 cmap 6385  ℕ_∞xnninf 6768

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1o 6163  df-2o 6164  df-map 6387  df-nninf 6770

This theorem is referenced by:  nninfsel  11553