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Theorem nninfsellemeqinf 13537
 Description: Lemma for nninfsel 13538. (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 13534 . . . . . 6 𝐸:(2o𝑚)⟶ℕ
32a1i 9 . . . . 5 (𝜑𝐸:(2o𝑚)⟶ℕ)
4 nninfsel.q . . . . 5 (𝜑𝑄 ∈ (2o𝑚))
53, 4ffvelrnd 5596 . . . 4 (𝜑 → (𝐸𝑄) ∈ ℕ)
6 nninff 13524 . . . 4 ((𝐸𝑄) ∈ ℕ → (𝐸𝑄):ω⟶2o)
75, 6syl 14 . . 3 (𝜑 → (𝐸𝑄):ω⟶2o)
87ffnd 5313 . 2 (𝜑 → (𝐸𝑄) Fn ω)
9 1onn 6456 . . . . 5 1o ∈ ω
10 fnconstg 5360 . . . . 5 (1o ∈ ω → (ω × {1o}) Fn ω)
119, 10ax-mp 5 . . . 4 (ω × {1o}) Fn ω
12 fconstmpt 4626 . . . . 5 (ω × {1o}) = (𝑖 ∈ ω ↦ 1o)
1312fneq1i 5257 . . . 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 2130 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑖𝑗𝑖𝑘))
1716ifbid 3522 . . . . . . . . 9 (𝑗 = 𝑘 → if(𝑖𝑗, 1o, ∅) = if(𝑖𝑘, 1o, ∅))
1817mpteq2dv 4051 . . . . . . . 8 (𝑗 = 𝑘 → (𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅)))
1918fveq2d 5465 . . . . . . 7 (𝑗 = 𝑘 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))))
2019eqeq1d 2163 . . . . . 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 13536 . . . . . . . 8 ((𝜑𝑗 ∈ ω) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o)
2625ralrimiva 2527 . . . . . . 7 (𝜑 → ∀𝑗 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑗, 1o, ∅))) = 1o)
2726ad2antrr 480 . . . . . 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 4548 . . . . . . . 8 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
3029ad2antlr 481 . . . . . . 7 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → suc 𝑗 ∈ ω)
31 elnn 4559 . . . . . . 7 ((𝑘 ∈ suc 𝑗 ∧ suc 𝑗 ∈ ω) → 𝑘 ∈ ω)
3228, 30, 31syl2anc 409 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → 𝑘 ∈ ω)
3320, 27, 32rspcdva 2818 . . . . 5 (((𝜑𝑗 ∈ ω) ∧ 𝑘 ∈ suc 𝑗) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o)
3433ralrimiva 2527 . . . 4 ((𝜑𝑗 ∈ ω) → ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o)
3534iftrued 3508 . . 3 ((𝜑𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) = 1o)
36 omex 4546 . . . . . . 7 ω ∈ V
3736mptex 5686 . . . . . 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 5460 . . . . . . . . . 10 (𝑞 = 𝑄 → (𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))))
4039eqeq1d 2163 . . . . . . . . 9 (𝑞 = 𝑄 → ((𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
4140ralbidv 2454 . . . . . . . 8 (𝑞 = 𝑄 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o ↔ ∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
4241ifbid 3522 . . . . . . 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 4051 . . . . . 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 5537 . . . . 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 409 . . . 4 ((𝜑𝑗 ∈ ω) → (𝐸𝑄) = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)))
46 suceq 4357 . . . . . . 7 (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
4746adantl 275 . . . . . 6 (((𝜑𝑗 ∈ ω) ∧ 𝑛 = 𝑗) → suc 𝑛 = suc 𝑗)
4847raleqdv 2655 . . . . 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 3522 . . . 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 2245 . . . 4 ((𝜑𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ ω)
5145, 49, 24, 50fvmptd 5542 . . 3 ((𝜑𝑗 ∈ ω) → ((𝐸𝑄)‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅))
52 eqidd 2155 . . . . . 6 (𝑖 = 𝑗 → 1o = 1o)
53 eqid 2154 . . . . . 6 (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
5452, 53fvmptg 5537 . . . . 5 ((𝑗 ∈ ω ∧ 1o ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
559, 54mpan2 422 . . . 4 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
5655adantl 275 . . 3 ((𝜑𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
5735, 51, 563eqtr4d 2197 . 2 ((𝜑𝑗 ∈ ω) → ((𝐸𝑄)‘𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
588, 15, 57eqfnfvd 5561 1 (𝜑 → (𝐸𝑄) = (𝑖 ∈ ω ↦ 1o))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 2125  ∀wral 2432  Vcvv 2709  ∅c0 3390  ifcif 3501  {csn 3556   ↦ cmpt 4021  suc csuc 4320  ωcom 4543   × cxp 4577   Fn wfn 5158  ⟶wf 5159  ‘cfv 5163  (class class class)co 5814  1oc1o 6346  2oc2o 6347   ↑𝑚 cmap 6582  ℕ∞xnninf 7049 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-iord 4321  df-on 4323  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1o 6353  df-2o 6354  df-map 6584  df-nninf 7050 This theorem is referenced by:  nninfsel  13538
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