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Theorem nninfsellemeqinf 12796
Description: Lemma for nninfsel 12797. (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 12793 . . . . . 6 𝐸:(2o𝑚)⟶ℕ
32a1i 9 . . . . 5 (𝜑𝐸:(2o𝑚)⟶ℕ)
4 nninfsel.q . . . . 5 (𝜑𝑄 ∈ (2o𝑚))
53, 4ffvelrnd 5488 . . . 4 (𝜑 → (𝐸𝑄) ∈ ℕ)
6 nninff 12782 . . . 4 ((𝐸𝑄) ∈ ℕ → (𝐸𝑄):ω⟶2o)
75, 6syl 14 . . 3 (𝜑 → (𝐸𝑄):ω⟶2o)
87ffnd 5209 . 2 (𝜑 → (𝐸𝑄) Fn ω)
9 1onn 6346 . . . . 5 1o ∈ ω
10 fnconstg 5256 . . . . 5 (1o ∈ ω → (ω × {1o}) Fn ω)
119, 10ax-mp 7 . . . 4 (ω × {1o}) Fn ω
12 fconstmpt 4524 . . . . 5 (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
1312fneq1i 5153 . . . 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 1659 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑖𝑗𝑖𝑘))
1716ifbid 3440 . . . . . . . . 9 (𝑗 = 𝑘 → if(𝑖𝑗, 1o, ∅) = if(𝑖𝑘, 1o, ∅))
1817mpteq2dv 3959 . . . . . . . 8 (𝑗 = 𝑘 → (𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅)))
1918fveq2d 5357 . . . . . . 7 (𝑗 = 𝑘 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))))
2019eqeq1d 2108 . . . . . 6 (𝑗 = 𝑘 → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
214adantr 272 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → 𝑄 ∈ (2o𝑚))
22 nninfsel.1 . . . . . . . . . 10 (𝜑 → (𝑄‘(𝐸𝑄)) = 1o)
2322adantr 272 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → (𝑄‘(𝐸𝑄)) = 1o)
24 simpr 109 . . . . . . . . 9 ((𝜑𝑗 ∈ ω) → 𝑗 ∈ ω)
251, 21, 23, 24nninfsellemqall 12795 . . . . . . . 8 ((𝜑𝑗 ∈ ω) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o)
2625ralrimiva 2464 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o)
2726ad2antrr 475 . . . . . 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 4447 . . . . . . . 8 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
3029ad2antlr 476 . . . . . . 7 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → suc 𝑗 ∈ ω)
31 elnn 4457 . . . . . . 7 ((𝑘 ∈ suc 𝑗 ∧ suc 𝑗 ∈ ω) → 𝑘 ∈ ω)
3228, 30, 31syl2anc 406 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ ω)
3320, 27, 32rspcdva 2749 . . . . 5 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o)
3433ralrimiva 2464 . . . 4 ((𝜑𝑗 ∈ ω) → ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o)
3534iftrued 3428 . . 3 ((𝜑𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) = 1o)
36 omex 4445 . . . . . . 7 ω ∈ V
3736mptex 5578 . . . . . 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 5352 . . . . . . . . . 10 (𝑞 = 𝑄 → (𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))))
4039eqeq1d 2108 . . . . . . . . 9 (𝑞 = 𝑄 → ((𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
4140ralbidv 2396 . . . . . . . 8 (𝑞 = 𝑄 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o ↔ ∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
4241ifbid 3440 . . . . . . 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 3959 . . . . . 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 5429 . . . . 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 406 . . . 4 ((𝜑𝑗 ∈ ω) → (𝐸𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)))
46 suceq 4262 . . . . . . 7 (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
4746adantl 273 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → suc 𝑛 = suc 𝑗)
4847raleqdv 2590 . . . . 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 3440 . . . 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 2190 . . . 4 ((𝜑𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ ω)
5145, 49, 24, 50fvmptd 5434 . . 3 ((𝜑𝑗 ∈ ω) → ((𝐸𝑄)‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅))
52 eqidd 2101 . . . . . 6 (𝑖 = 𝑗 → 1o = 1o)
53 eqid 2100 . . . . . 6 (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
5452, 53fvmptg 5429 . . . . 5 ((𝑗 ∈ ω ∧ 1o ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
559, 54mpan2 419 . . . 4 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
5655adantl 273 . . 3 ((𝜑𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
5735, 51, 563eqtr4d 2142 . 2 ((𝜑𝑗 ∈ ω) → ((𝐸𝑄)‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
588, 15, 57eqfnfvd 5453 1 (𝜑 → (𝐸𝑄) = (𝑖 ∈ ω ↦ 1o))
Colors of variables: wff set class
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  1oc1o 6236  2oc2o 6237  𝑚 cmap 6472  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
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