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Theorem nninfself 11551
Description: Domain and range of the selection function for . (Contributed by Jim Kingdon, 6-Aug-2022.)
Hypothesis
Ref Expression
nninfsel.e 𝐸 = (𝑞 ∈ (2𝑜𝑚) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)))
Assertion
Ref Expression
nninfself 𝐸:(2𝑜𝑚)⟶ℕ
Distinct variable groups:   𝑖,𝑘,𝑛   𝑘,𝑞,𝑛
Allowed substitution hints:   𝐸(𝑖,𝑘,𝑛,𝑞)

Proof of Theorem nninfself
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nninfsel.e . 2 𝐸 = (𝑞 ∈ (2𝑜𝑚) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)))
2 nninfsellemcl 11549 . . . . 5 ((𝑞 ∈ (2𝑜𝑚) ∧ 𝑛 ∈ ω) → if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜)
3 eqid 2088 . . . . 5 (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, ∅))
42, 3fmptd 5436 . . . 4 (𝑞 ∈ (2𝑜𝑚) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)):ω⟶2𝑜)
5 2onn 6260 . . . . . 6 2𝑜 ∈ ω
65a1i 9 . . . . 5 (𝑞 ∈ (2𝑜𝑚) → 2𝑜 ∈ ω)
7 omex 4398 . . . . . 6 ω ∈ V
87a1i 9 . . . . 5 (𝑞 ∈ (2𝑜𝑚) → ω ∈ V)
96, 8elmapd 6399 . . . 4 (𝑞 ∈ (2𝑜𝑚) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ (2𝑜𝑚 ω) ↔ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)):ω⟶2𝑜))
104, 9mpbird 165 . . 3 (𝑞 ∈ (2𝑜𝑚) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ (2𝑜𝑚 ω))
11 nninfsellemsuc 11550 . . . . 5 ((𝑞 ∈ (2𝑜𝑚) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ⊆ if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
12 peano2 4400 . . . . . 6 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
13 nninfsellemcl 11549 . . . . . . 7 ((𝑞 ∈ (2𝑜𝑚) ∧ suc 𝑗 ∈ ω) → if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜)
1412, 13sylan2 280 . . . . . 6 ((𝑞 ∈ (2𝑜𝑚) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜)
15 suceq 4220 . . . . . . . . 9 (𝑛 = suc 𝑗 → suc 𝑛 = suc suc 𝑗)
1615raleqdv 2568 . . . . . . . 8 (𝑛 = suc 𝑗 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜))
1716ifbid 3408 . . . . . . 7 (𝑛 = suc 𝑗 → if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) = if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
1817, 3fvmptg 5364 . . . . . 6 ((suc 𝑗 ∈ ω ∧ if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) = if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
1912, 14, 18syl2an2 561 . . . . 5 ((𝑞 ∈ (2𝑜𝑚) ∧ 𝑗 ∈ ω) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) = if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
20 simpr 108 . . . . . 6 ((𝑞 ∈ (2𝑜𝑚) ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
21 nninfsellemcl 11549 . . . . . 6 ((𝑞 ∈ (2𝑜𝑚) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜)
22 suceq 4220 . . . . . . . . 9 (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
2322raleqdv 2568 . . . . . . . 8 (𝑛 = 𝑗 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜))
2423ifbid 3408 . . . . . . 7 (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, ∅))
2524, 3fvmptg 5364 . . . . . 6 ((𝑗 ∈ ω ∧ if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅) ∈ 2𝑜) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
2620, 21, 25syl2anc 403 . . . . 5 ((𝑞 ∈ (2𝑜𝑚) ∧ 𝑗 ∈ ω) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗) = if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))
2711, 19, 263sstr4d 3067 . . . 4 ((𝑞 ∈ (2𝑜𝑚) ∧ 𝑗 ∈ ω) → ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) ⊆ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗))
2827ralrimiva 2446 . . 3 (𝑞 ∈ (2𝑜𝑚) → ∀𝑗 ∈ ω ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) ⊆ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗))
29 fveq1 5288 . . . . . 6 (𝑓 = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) → (𝑓‘suc 𝑗) = ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗))
30 fveq1 5288 . . . . . 6 (𝑓 = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) → (𝑓𝑗) = ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗))
3129, 30sseq12d 3053 . . . . 5 (𝑓 = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) ⊆ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗)))
3231ralbidv 2380 . . . 4 (𝑓 = (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) ⊆ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗)))
33 df-nninf 6770 . . . 4 = {𝑓 ∈ (2𝑜𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
3432, 33elrab2 2772 . . 3 ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ ℕ ↔ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ (2𝑜𝑚 ω) ∧ ∀𝑗 ∈ ω ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘suc 𝑗) ⊆ ((𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅))‘𝑗)))
3510, 28, 34sylanbrc 408 . 2 (𝑞 ∈ (2𝑜𝑚) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1𝑜, ∅))) = 1𝑜, 1𝑜, ∅)) ∈ ℕ)
361, 35fmpti 5435 1 𝐸:(2𝑜𝑚)⟶ℕ
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wcel 1438  wral 2359  Vcvv 2619  wss 2997  c0 3284  ifcif 3389  cmpt 3891  suc csuc 4183  ωcom 4395  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-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-rab 2368  df-v 2621  df-sbc 2839  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-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-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:  nninfsellemeq  11552  nninfsellemeqinf  11554  nninfomnilem  11556
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