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Theorem nninfsellemeqinf 11552
Description: Lemma for nninfsel 11553. (Contributed by Jim Kingdon, 9-Aug-2022.)
Hypotheses
Ref Expression
nninfsel.e 𝐸 = (𝑞 ∈ (2𝑜𝑚) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)))
nninfsel.q (𝜑𝑄 ∈ (2𝑜𝑚))
nninfsel.1 (𝜑 → (𝑄‘(𝐸𝑄)) = 1𝑜)
Assertion
Ref Expression
nninfsellemeqinf (𝜑 → (𝐸𝑄) = (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.
StepHypRef Expression
1 nninfsel.e . . . . . . 7 𝐸 = (𝑞 ∈ (2𝑜𝑚) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)))
21nninfself 11549 . . . . . 6 𝐸:(2𝑜𝑚)⟶ℕ
32a1i 9 . . . . 5 (𝜑𝐸:(2𝑜𝑚)⟶ℕ)
4 nninfsel.q . . . . 5 (𝜑𝑄 ∈ (2𝑜𝑚))
53, 4ffvelrnd 5419 . . . 4 (𝜑 → (𝐸𝑄) ∈ ℕ)
6 nninff 11538 . . . 4 ((𝐸𝑄) ∈ ℕ → (𝐸𝑄):ω⟶2𝑜)
75, 6syl 14 . . 3 (𝜑 → (𝐸𝑄):ω⟶2𝑜)
87ffnd 5148 . 2 (𝜑 → (𝐸𝑄) Fn ω)
9 1onn 6259 . . . . 5 1𝑜 ∈ ω
10 fnconstg 5192 . . . . 5 (1𝑜 ∈ ω → (ω × {1𝑜}) Fn ω)
119, 10ax-mp 7 . . . 4 (ω × {1𝑜}) Fn ω
12 fconstmpt 4473 . . . . 5 (ω × {1𝑜}) = (𝑖 ∈ ω ↦ 1𝑜)
1312fneq1i 5094 . . . 4 ((ω × {1𝑜}) Fn ω ↔ (𝑖 ∈ ω ↦ 1𝑜) Fn ω)
1411, 13mpbi 143 . . 3 (𝑖 ∈ ω ↦ 1𝑜) Fn ω
1514a1i 9 . 2 (𝜑 → (𝑖 ∈ ω ↦ 1𝑜) Fn ω)
16 elequ2 1648 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑖𝑗𝑖𝑘))
1716ifbid 3408 . . . . . . . . 9 (𝑗 = 𝑘 → if(𝑖𝑗, 1𝑜, ∅) = if(𝑖𝑘, 1𝑜, ∅))
1817mpteq2dv 3921 . . . . . . . 8 (𝑗 = 𝑘 → (𝑖 ∈ ω ↦ if(𝑖𝑗, 1𝑜, ∅)) = (𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅)))
1918fveq2d 5293 . . . . . . 7 (𝑗 = 𝑘 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1𝑜, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))))
2019eqeq1d 2096 . . . . . 6 (𝑗 = 𝑘 → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1𝑜, ∅))) = 1𝑜 ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜))
214adantr 270 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → 𝑄 ∈ (2𝑜𝑚))
22 nninfsel.1 . . . . . . . . . 10 (𝜑 → (𝑄‘(𝐸𝑄)) = 1𝑜)
2322adantr 270 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → (𝑄‘(𝐸𝑄)) = 1𝑜)
24 simpr 108 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → 𝑗 ∈ ω)
251, 21, 23, 24nninfsellemqall 11551 . . . . . . . 8 ((𝜑𝑗 ∈ ω) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1𝑜, ∅))) = 1𝑜)
2625ralrimiva 2446 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1𝑜, ∅))) = 1𝑜)
2726ad2antrr 472 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1𝑜, ∅))) = 1𝑜)
28 simpr 108 . . . . . . 7 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ suc 𝑗)
29 peano2 4400 . . . . . . . 8 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
3029ad2antlr 473 . . . . . . 7 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → suc 𝑗 ∈ ω)
31 elnn 4410 . . . . . . 7 ((𝑘 ∈ suc 𝑗 ∧ suc 𝑗 ∈ ω) → 𝑘 ∈ ω)
3228, 30, 31syl2anc 403 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ ω)
3320, 27, 32rspcdva 2727 . . . . 5 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜)
3433ralrimiva 2446 . . . 4 ((𝜑𝑗 ∈ ω) → ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜)
3534iftrued 3396 . . 3 ((𝜑𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) = 1𝑜)
36 omex 4398 . . . . . . 7 ω ∈ V
3736mptex 5505 . . . . . 6 (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ V
3837a1i 9 . . . . 5 ((𝜑𝑗 ∈ ω) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ V)
39 fveq1 5288 . . . . . . . . . 10 (𝑞 = 𝑄 → (𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))))
4039eqeq1d 2096 . . . . . . . . 9 (𝑞 = 𝑄 → ((𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜 ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜))
4140ralbidv 2380 . . . . . . . 8 (𝑞 = 𝑄 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜))
4241ifbid 3408 . . . . . . 7 (𝑞 = 𝑄 → if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) = if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
4342mpteq2dv 3921 . . . . . 6 (𝑞 = 𝑄 → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)))
4443, 1fvmptg 5364 . . . . 5 ((𝑄 ∈ (2𝑜𝑚) ∧ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ V) → (𝐸𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)))
4521, 38, 44syl2anc 403 . . . 4 ((𝜑𝑗 ∈ ω) → (𝐸𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)))
46 suceq 4220 . . . . . . 7 (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
4746adantl 271 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → suc 𝑛 = suc 𝑗)
4847raleqdv 2568 . . . . 5 (((𝜑𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → (∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜))
4948ifbid 3408 . . . 4 (((𝜑𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) = if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
5035, 9syl6eqel 2178 . . . 4 ((𝜑𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ ω)
5145, 49, 24, 50fvmptd 5369 . . 3 ((𝜑𝑗 ∈ ω) → ((𝐸𝑄)‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
52 eqidd 2089 . . . . . 6 (𝑖 = 𝑗 → 1𝑜 = 1𝑜)
53 eqid 2088 . . . . . 6 (𝑖 ∈ ω ↦ 1𝑜) = (𝑖 ∈ ω ↦ 1𝑜)
5452, 53fvmptg 5364 . . . . 5 ((𝑗 ∈ ω ∧ 1𝑜 ∈ ω) → ((𝑖 ∈ ω ↦ 1𝑜)‘𝑗) = 1𝑜)
559, 54mpan2 416 . . . 4 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1𝑜)‘𝑗) = 1𝑜)
5655adantl 271 . . 3 ((𝜑𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ 1𝑜)‘𝑗) = 1𝑜)
5735, 51, 563eqtr4d 2130 . 2 ((𝜑𝑗 ∈ ω) → ((𝐸𝑄)‘𝑗) = ((𝑖 ∈ ω ↦ 1𝑜)‘𝑗))
588, 15, 57eqfnfvd 5384 1 (𝜑 → (𝐸𝑄) = (𝑖 ∈ ω ↦ 1𝑜))
Colors of variables: wff set class
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  1𝑜c1o 6156  2𝑜c2o 6157  𝑚 cmap 6385  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
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