ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  hashinfom GIF version

Theorem hashinfom 10729
Description: The value of the function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.)
Assertion
Ref Expression
hashinfom (ω ≼ 𝐴 → (♯‘𝐴) = +∞)

Proof of Theorem hashinfom
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ihash 10727 . . . . 5 ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
21fveq1i 5511 . . . 4 (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))‘𝐴)
3 funmpt 5249 . . . . 5 Fun (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
4 funrel 5228 . . . . . . 7 (Fun (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) → Rel (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
53, 4ax-mp 5 . . . . . 6 Rel (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
6 peano1 4589 . . . . . . 7 ∅ ∈ ω
7 reldom 6738 . . . . . . . . . 10 Rel ≼
87brrelex2i 4666 . . . . . . . . 9 (ω ≼ 𝐴𝐴 ∈ V)
9 hashinfuni 10728 . . . . . . . . . 10 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = ω)
10 omex 4588 . . . . . . . . . 10 ω ∈ V
119, 10eqeltrdi 2268 . . . . . . . . 9 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ V)
12 breq2 4004 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
1312rabbidv 2726 . . . . . . . . . . 11 (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1413unieqd 3818 . . . . . . . . . 10 (𝑥 = 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
15 eqid 2177 . . . . . . . . . 10 (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) = (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
1614, 15fvmptg 5587 . . . . . . . . 9 ((𝐴 ∈ V ∧ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ V) → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
178, 11, 16syl2anc 411 . . . . . . . 8 (ω ≼ 𝐴 → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1817, 9eqtrd 2210 . . . . . . 7 (ω ≼ 𝐴 → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = ω)
196, 18eleqtrrid 2267 . . . . . 6 (ω ≼ 𝐴 → ∅ ∈ ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴))
20 relelfvdm 5542 . . . . . 6 ((Rel (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) ∧ ∅ ∈ ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴)) → 𝐴 ∈ dom (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
215, 19, 20sylancr 414 . . . . 5 (ω ≼ 𝐴𝐴 ∈ dom (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
22 fvco 5581 . . . . 5 ((Fun (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) ∧ 𝐴 ∈ dom (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})) → (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴)))
233, 21, 22sylancr 414 . . . 4 (ω ≼ 𝐴 → (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴)))
242, 23eqtrid 2222 . . 3 (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴)))
2518fveq2d 5514 . . 3 (ω ≼ 𝐴 → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴)) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω))
2624, 25eqtrd 2210 . 2 (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω))
27 pnfxr 7987 . . 3 +∞ ∈ ℝ*
28 ordom 4602 . . . . 5 Ord ω
29 ordirr 4537 . . . . 5 (Ord ω → ¬ ω ∈ ω)
3028, 29ax-mp 5 . . . 4 ¬ ω ∈ ω
31 zex 9238 . . . . . . . . . 10 ℤ ∈ V
3231mptex 5737 . . . . . . . . 9 (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
33 vex 2740 . . . . . . . . 9 𝑧 ∈ V
3432, 33fvex 5530 . . . . . . . 8 ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
3534ax-gen 1449 . . . . . . 7 𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
36 0z 9240 . . . . . . 7 0 ∈ ℤ
37 frecfnom 6395 . . . . . . 7 ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3835, 36, 37mp2an 426 . . . . . 6 frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
39 fndm 5310 . . . . . 6 (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω → dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) = ω)
4038, 39ax-mp 5 . . . . 5 dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) = ω
4140eleq2i 2244 . . . 4 (ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ↔ ω ∈ ω)
4230, 41mtbir 671 . . 3 ¬ ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
43 fsnunfv 5712 . . 3 ((ω ∈ V ∧ +∞ ∈ ℝ* ∧ ¬ ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)) → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω) = +∞)
4410, 27, 42, 43mp3an 1337 . 2 ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω) = +∞
4526, 44eqtrdi 2226 1 (ω ≼ 𝐴 → (♯‘𝐴) = +∞)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1351   = wceq 1353  wcel 2148  {crab 2459  Vcvv 2737  cun 3127  c0 3422  {csn 3591  cop 3594   cuni 3807   class class class wbr 4000  cmpt 4061  Ord word 4358  ωcom 4585  dom cdm 4622  ccom 4626  Rel wrel 4627  Fun wfun 5205   Fn wfn 5206  cfv 5211  (class class class)co 5868  freccfrec 6384  cdom 6732  0cc0 7789  1c1 7790   + caddc 7792  +∞cpnf 7966  *cxr 7968  cz 9229  chash 10726
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886  ax-rnegex 7898
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-recs 6299  df-frec 6385  df-dom 6735  df-pnf 7971  df-xr 7973  df-neg 8108  df-z 9230  df-ihash 10727
This theorem is referenced by:  filtinf  10742
  Copyright terms: Public domain W3C validator