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Theorem nninfsellemeqinf 15244
Description: Lemma for nninfsel 15245. (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 15241 . . . . . 6 𝐸:(2o𝑚)⟶ℕ
32a1i 9 . . . . 5 (𝜑𝐸:(2o𝑚)⟶ℕ)
4 nninfsel.q . . . . 5 (𝜑𝑄 ∈ (2o𝑚))
53, 4ffvelcdmd 5673 . . . 4 (𝜑 → (𝐸𝑄) ∈ ℕ)
6 nninff 7152 . . . 4 ((𝐸𝑄) ∈ ℕ → (𝐸𝑄):ω⟶2o)
75, 6syl 14 . . 3 (𝜑 → (𝐸𝑄):ω⟶2o)
87ffnd 5385 . 2 (𝜑 → (𝐸𝑄) Fn ω)
9 1onn 6546 . . . . 5 1o ∈ ω
10 fnconstg 5432 . . . . 5 (1o ∈ ω → (ω × {1o}) Fn ω)
119, 10ax-mp 5 . . . 4 (ω × {1o}) Fn ω
12 fconstmpt 4691 . . . . 5 (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
1312fneq1i 5329 . . . 4 ((ω × {1o}) Fn ω ↔ (𝑖 ∈ ω ↦ 1o) Fn ω)
1411, 13mpbi 145 . . 3 (𝑖 ∈ ω ↦ 1o) Fn ω
1514a1i 9 . 2 (𝜑 → (𝑖 ∈ ω ↦ 1o) Fn ω)
16 elequ2 2165 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑖𝑗𝑖𝑘))
1716ifbid 3570 . . . . . . . . 9 (𝑗 = 𝑘 → if(𝑖𝑗, 1o, ∅) = if(𝑖𝑘, 1o, ∅))
1817mpteq2dv 4109 . . . . . . . 8 (𝑗 = 𝑘 → (𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅)))
1918fveq2d 5538 . . . . . . 7 (𝑗 = 𝑘 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))))
2019eqeq1d 2198 . . . . . 6 (𝑗 = 𝑘 → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
214adantr 276 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → 𝑄 ∈ (2o𝑚))
22 nninfsel.1 . . . . . . . . . 10 (𝜑 → (𝑄‘(𝐸𝑄)) = 1o)
2322adantr 276 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → (𝑄‘(𝐸𝑄)) = 1o)
24 simpr 110 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → 𝑗 ∈ ω)
251, 21, 23, 24nninfsellemqall 15243 . . . . . . . 8 ((𝜑𝑗 ∈ ω) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o)
2625ralrimiva 2563 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o)
2726ad2antrr 488 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o)
28 simpr 110 . . . . . . 7 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ suc 𝑗)
29 peano2 4612 . . . . . . . 8 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
3029ad2antlr 489 . . . . . . 7 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → suc 𝑗 ∈ ω)
31 elnn 4623 . . . . . . 7 ((𝑘 ∈ suc 𝑗 ∧ suc 𝑗 ∈ ω) → 𝑘 ∈ ω)
3228, 30, 31syl2anc 411 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ ω)
3320, 27, 32rspcdva 2861 . . . . 5 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o)
3433ralrimiva 2563 . . . 4 ((𝜑𝑗 ∈ ω) → ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o)
3534iftrued 3556 . . 3 ((𝜑𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) = 1o)
36 omex 4610 . . . . . . 7 ω ∈ V
3736mptex 5763 . . . . . 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 5533 . . . . . . . . . 10 (𝑞 = 𝑄 → (𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))))
4039eqeq1d 2198 . . . . . . . . 9 (𝑞 = 𝑄 → ((𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
4140ralbidv 2490 . . . . . . . 8 (𝑞 = 𝑄 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o ↔ ∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
4241ifbid 3570 . . . . . . 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 4109 . . . . . 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 5613 . . . . 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 411 . . . 4 ((𝜑𝑗 ∈ ω) → (𝐸𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)))
46 suceq 4420 . . . . . . 7 (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
4746adantl 277 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → suc 𝑛 = suc 𝑗)
4847raleqdv 2692 . . . . 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 3570 . . . 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, 9eqeltrdi 2280 . . . 4 ((𝜑𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ ω)
5145, 49, 24, 50fvmptd 5618 . . 3 ((𝜑𝑗 ∈ ω) → ((𝐸𝑄)‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅))
52 eqidd 2190 . . . . . 6 (𝑖 = 𝑗 → 1o = 1o)
53 eqid 2189 . . . . . 6 (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
5452, 53fvmptg 5613 . . . . 5 ((𝑗 ∈ ω ∧ 1o ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
559, 54mpan2 425 . . . 4 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
5655adantl 277 . . 3 ((𝜑𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
5735, 51, 563eqtr4d 2232 . 2 ((𝜑𝑗 ∈ ω) → ((𝐸𝑄)‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
588, 15, 57eqfnfvd 5637 1 (𝜑 → (𝐸𝑄) = (𝑖 ∈ ω ↦ 1o))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wral 2468  Vcvv 2752  c0 3437  ifcif 3549  {csn 3607  cmpt 4079  suc csuc 4383  ωcom 4607   × cxp 4642   Fn wfn 5230  wf 5231  cfv 5235  (class class class)co 5897  1oc1o 6435  2oc2o 6436  𝑚 cmap 6675  xnninf 7149
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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1o 6442  df-2o 6443  df-map 6677  df-nninf 7150
This theorem is referenced by:  nninfsel  15245
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