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Theorem nninfsellemeqinf 13139
Description: Lemma for nninfsel 13140. (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 13136 . . . . . 6 𝐸:(2o𝑚)⟶ℕ
32a1i 9 . . . . 5 (𝜑𝐸:(2o𝑚)⟶ℕ)
4 nninfsel.q . . . . 5 (𝜑𝑄 ∈ (2o𝑚))
53, 4ffvelrnd 5524 . . . 4 (𝜑 → (𝐸𝑄) ∈ ℕ)
6 nninff 13125 . . . 4 ((𝐸𝑄) ∈ ℕ → (𝐸𝑄):ω⟶2o)
75, 6syl 14 . . 3 (𝜑 → (𝐸𝑄):ω⟶2o)
87ffnd 5243 . 2 (𝜑 → (𝐸𝑄) Fn ω)
9 1onn 6384 . . . . 5 1o ∈ ω
10 fnconstg 5290 . . . . 5 (1o ∈ ω → (ω × {1o}) Fn ω)
119, 10ax-mp 5 . . . 4 (ω × {1o}) Fn ω
12 fconstmpt 4556 . . . . 5 (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
1312fneq1i 5187 . . . 4 ((ω × {1o}) Fn ω ↔ (𝑖 ∈ ω ↦ 1o) Fn ω)
1411, 13mpbi 144 . . 3 (𝑖 ∈ ω ↦ 1o) Fn ω
1514a1i 9 . 2 (𝜑 → (𝑖 ∈ ω ↦ 1o) Fn ω)
16 elequ2 1676 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑖𝑗𝑖𝑘))
1716ifbid 3463 . . . . . . . . 9 (𝑗 = 𝑘 → if(𝑖𝑗, 1o, ∅) = if(𝑖𝑘, 1o, ∅))
1817mpteq2dv 3989 . . . . . . . 8 (𝑗 = 𝑘 → (𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅)))
1918fveq2d 5393 . . . . . . 7 (𝑗 = 𝑘 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))))
2019eqeq1d 2126 . . . . . 6 (𝑗 = 𝑘 → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
214adantr 274 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → 𝑄 ∈ (2o𝑚))
22 nninfsel.1 . . . . . . . . . 10 (𝜑 → (𝑄‘(𝐸𝑄)) = 1o)
2322adantr 274 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → (𝑄‘(𝐸𝑄)) = 1o)
24 simpr 109 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → 𝑗 ∈ ω)
251, 21, 23, 24nninfsellemqall 13138 . . . . . . . 8 ((𝜑𝑗 ∈ ω) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o)
2625ralrimiva 2482 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o)
2726ad2antrr 479 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o)
28 simpr 109 . . . . . . 7 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ suc 𝑗)
29 peano2 4479 . . . . . . . 8 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
3029ad2antlr 480 . . . . . . 7 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → suc 𝑗 ∈ ω)
31 elnn 4489 . . . . . . 7 ((𝑘 ∈ suc 𝑗 ∧ suc 𝑗 ∈ ω) → 𝑘 ∈ ω)
3228, 30, 31syl2anc 408 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ ω)
3320, 27, 32rspcdva 2768 . . . . 5 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o)
3433ralrimiva 2482 . . . 4 ((𝜑𝑗 ∈ ω) → ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o)
3534iftrued 3451 . . 3 ((𝜑𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) = 1o)
36 omex 4477 . . . . . . 7 ω ∈ V
3736mptex 5614 . . . . . 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 5388 . . . . . . . . . 10 (𝑞 = 𝑄 → (𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))))
4039eqeq1d 2126 . . . . . . . . 9 (𝑞 = 𝑄 → ((𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
4140ralbidv 2414 . . . . . . . 8 (𝑞 = 𝑄 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o ↔ ∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
4241ifbid 3463 . . . . . . 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 3989 . . . . . 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 5465 . . . . 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 408 . . . 4 ((𝜑𝑗 ∈ ω) → (𝐸𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)))
46 suceq 4294 . . . . . . 7 (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
4746adantl 275 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → suc 𝑛 = suc 𝑗)
4847raleqdv 2609 . . . . 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 3463 . . . 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 2208 . . . 4 ((𝜑𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ ω)
5145, 49, 24, 50fvmptd 5470 . . 3 ((𝜑𝑗 ∈ ω) → ((𝐸𝑄)‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅))
52 eqidd 2118 . . . . . 6 (𝑖 = 𝑗 → 1o = 1o)
53 eqid 2117 . . . . . 6 (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
5452, 53fvmptg 5465 . . . . 5 ((𝑗 ∈ ω ∧ 1o ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
559, 54mpan2 421 . . . 4 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
5655adantl 275 . . 3 ((𝜑𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
5735, 51, 563eqtr4d 2160 . 2 ((𝜑𝑗 ∈ ω) → ((𝐸𝑄)‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
588, 15, 57eqfnfvd 5489 1 (𝜑 → (𝐸𝑄) = (𝑖 ∈ ω ↦ 1o))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  wral 2393  Vcvv 2660  c0 3333  ifcif 3444  {csn 3497  cmpt 3959  suc csuc 4257  ωcom 4474   × cxp 4507   Fn wfn 5088  wf 5089  cfv 5093  (class class class)co 5742  1oc1o 6274  2oc2o 6275  𝑚 cmap 6510  xnninf 6973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1o 6281  df-2o 6282  df-map 6512  df-nninf 6975
This theorem is referenced by:  nninfsel  13140
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