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Theorem nninfself 14303
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 14301 . . . . 5 ((𝑞 ∈ (2o𝑚) ∧ 𝑛 ∈ ω) → if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
3 eqid 2175 . . . . 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 5662 . . . 4 (𝑞 ∈ (2o𝑚) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)):ω⟶2o)
5 2onn 6512 . . . . . 6 2o ∈ ω
65a1i 9 . . . . 5 (𝑞 ∈ (2o𝑚) → 2o ∈ ω)
7 omex 4586 . . . . . 6 ω ∈ V
87a1i 9 . . . . 5 (𝑞 ∈ (2o𝑚) → ω ∈ V)
96, 8elmapd 6652 . . . 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 167 . . 3 (𝑞 ∈ (2o𝑚) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)) ∈ (2o𝑚 ω))
11 nninfsellemsuc 14302 . . . . 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 4588 . . . . . 6 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
13 nninfsellemcl 14301 . . . . . . 7 ((𝑞 ∈ (2o𝑚) ∧ suc 𝑗 ∈ ω) → if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
1412, 13sylan2 286 . . . . . 6 ((𝑞 ∈ (2o𝑚) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
15 suceq 4396 . . . . . . . . 9 (𝑛 = suc 𝑗 → suc 𝑛 = suc suc 𝑗)
1615raleqdv 2676 . . . . . . . 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 3553 . . . . . . 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 5584 . . . . . 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 594 . . . . 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 110 . . . . . 6 ((𝑞 ∈ (2o𝑚) ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
21 nninfsellemcl 14301 . . . . . 6 ((𝑞 ∈ (2o𝑚) ∧ 𝑗 ∈ ω) → if(∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)
22 suceq 4396 . . . . . . . . 9 (𝑛 = 𝑗 → suc 𝑛 = suc 𝑗)
2322raleqdv 2676 . . . . . . . 8 (𝑛 = 𝑗 → (∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o ↔ ∀𝑘 ∈ suc 𝑗(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o))
2423ifbid 3553 . . . . . . 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 5584 . . . . . 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 411 . . . . 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 3198 . . . 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 2548 . . 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 5506 . . . . . 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 5506 . . . . . 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 3184 . . . . 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 2475 . . . 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 7109 . . . 4 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
3432, 33elrab2 2894 . . 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 417 . 2 (𝑞 ∈ (2o𝑚) → (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅)) ∈ ℕ)
361, 35fmpti 5660 1 𝐸:(2o𝑚)⟶ℕ
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wcel 2146  wral 2453  Vcvv 2735  wss 3127  c0 3420  ifcif 3532  cmpt 4059  suc csuc 4359  ωcom 4583  wf 5204  cfv 5208  (class class class)co 5865  1oc1o 6400  2oc2o 6401  𝑚 cmap 6638  xnninf 7108
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1o 6407  df-2o 6408  df-map 6640  df-nninf 7109
This theorem is referenced by:  nninfsellemeq  14304  nninfsellemeqinf  14306  nninfomnilem  14308
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