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Theorem hashinfom 10020
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 10018 . . . . 5 ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
21fveq1i 5253 . . . 4 (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))‘𝐴)
3 funmpt 5004 . . . . 5 Fun (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
4 funrel 4985 . . . . . . 7 (Fun (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) → Rel (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
53, 4ax-mp 7 . . . . . 6 Rel (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
6 peano1 4371 . . . . . . 7 ∅ ∈ ω
7 reldom 6391 . . . . . . . . . 10 Rel ≼
87brrelex2i 4440 . . . . . . . . 9 (ω ≼ 𝐴𝐴 ∈ V)
9 hashinfuni 10019 . . . . . . . . . 10 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = ω)
10 omex 4370 . . . . . . . . . 10 ω ∈ V
119, 10syl6eqel 2173 . . . . . . . . 9 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ V)
12 breq2 3815 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
1312rabbidv 2601 . . . . . . . . . . 11 (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1413unieqd 3638 . . . . . . . . . 10 (𝑥 = 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
15 eqid 2083 . . . . . . . . . 10 (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) = (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
1614, 15fvmptg 5324 . . . . . . . . 9 ((𝐴 ∈ V ∧ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ V) → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
178, 11, 16syl2anc 403 . . . . . . . 8 (ω ≼ 𝐴 → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1817, 9eqtrd 2115 . . . . . . 7 (ω ≼ 𝐴 → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = ω)
196, 18syl5eleqr 2172 . . . . . 6 (ω ≼ 𝐴 → ∅ ∈ ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴))
20 relelfvdm 5280 . . . . . 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 405 . . . . 5 (ω ≼ 𝐴𝐴 ∈ dom (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
22 fvco 5318 . . . . 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 405 . . . 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, 23syl5eq 2127 . . 3 (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴)))
2518fveq2d 5256 . . 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 2115 . 2 (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω))
27 pnfxr 7442 . . 3 +∞ ∈ ℝ*
28 ordom 4383 . . . . 5 Ord ω
29 ordirr 4320 . . . . 5 (Ord ω → ¬ ω ∈ ω)
3028, 29ax-mp 7 . . . 4 ¬ ω ∈ ω
31 zex 8654 . . . . . . . . . 10 ℤ ∈ V
3231mptex 5462 . . . . . . . . 9 (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
33 vex 2615 . . . . . . . . 9 𝑧 ∈ V
3432, 33fvex 5269 . . . . . . . 8 ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
3534ax-gen 1379 . . . . . . 7 𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
36 0z 8656 . . . . . . 7 0 ∈ ℤ
37 frecfnom 6097 . . . . . . 7 ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3835, 36, 37mp2an 417 . . . . . 6 frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
39 fndm 5065 . . . . . 6 (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω → dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) = ω)
4038, 39ax-mp 7 . . . . 5 dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) = ω
4140eleq2i 2149 . . . 4 (ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ↔ ω ∈ ω)
4230, 41mtbir 629 . . 3 ¬ ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
43 fsnunfv 5438 . . 3 ((ω ∈ V ∧ +∞ ∈ ℝ* ∧ ¬ ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)) → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω) = +∞)
4410, 27, 42, 43mp3an 1269 . 2 ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω) = +∞
4526, 44syl6eq 2131 1 (ω ≼ 𝐴 → (♯‘𝐴) = +∞)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1283   = wceq 1285  wcel 1434  {crab 2357  Vcvv 2612  cun 2982  c0 3269  {csn 3422  cop 3425   cuni 3627   class class class wbr 3811  cmpt 3865  Ord word 4152  ωcom 4367  dom cdm 4400  ccom 4404  Rel wrel 4405  Fun wfun 4962   Fn wfn 4963  cfv 4968  (class class class)co 5590  freccfrec 6086  cdom 6385  0cc0 7252  1c1 7253   + caddc 7255  +∞cpnf 7421  *cxr 7423  cz 8645  chash 10017
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7338  ax-resscn 7339  ax-1re 7341  ax-addrcl 7344  ax-rnegex 7356
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4406  df-rel 4407  df-cnv 4408  df-co 4409  df-dm 4410  df-rn 4411  df-res 4412  df-ima 4413  df-iota 4933  df-fun 4970  df-fn 4971  df-f 4972  df-f1 4973  df-fo 4974  df-f1o 4975  df-fv 4976  df-ov 5593  df-recs 6001  df-frec 6087  df-dom 6388  df-pnf 7426  df-xr 7428  df-neg 7558  df-z 8646  df-ihash 10018
This theorem is referenced by:  filtinf  10034
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