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Theorem hashinfom 10555
 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 10553 . . . . 5 ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
21fveq1i 5429 . . . 4 (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))‘𝐴)
3 funmpt 5168 . . . . 5 Fun (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
4 funrel 5147 . . . . . . 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 4515 . . . . . . 7 ∅ ∈ ω
7 reldom 6646 . . . . . . . . . 10 Rel ≼
87brrelex2i 4590 . . . . . . . . 9 (ω ≼ 𝐴𝐴 ∈ V)
9 hashinfuni 10554 . . . . . . . . . 10 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = ω)
10 omex 4514 . . . . . . . . . 10 ω ∈ V
119, 10eqeltrdi 2231 . . . . . . . . 9 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ V)
12 breq2 3940 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
1312rabbidv 2678 . . . . . . . . . . 11 (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1413unieqd 3754 . . . . . . . . . 10 (𝑥 = 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
15 eqid 2140 . . . . . . . . . 10 (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) = (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
1614, 15fvmptg 5504 . . . . . . . . 9 ((𝐴 ∈ V ∧ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ V) → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
178, 11, 16syl2anc 409 . . . . . . . 8 (ω ≼ 𝐴 → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1817, 9eqtrd 2173 . . . . . . 7 (ω ≼ 𝐴 → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = ω)
196, 18eleqtrrid 2230 . . . . . 6 (ω ≼ 𝐴 → ∅ ∈ ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴))
20 relelfvdm 5460 . . . . . 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 411 . . . . 5 (ω ≼ 𝐴𝐴 ∈ dom (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
22 fvco 5498 . . . . 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 411 . . . 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 2185 . . 3 (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴)))
2518fveq2d 5432 . . 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 2173 . 2 (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω))
27 pnfxr 7841 . . 3 +∞ ∈ ℝ*
28 ordom 4527 . . . . 5 Ord ω
29 ordirr 4464 . . . . 5 (Ord ω → ¬ ω ∈ ω)
3028, 29ax-mp 5 . . . 4 ¬ ω ∈ ω
31 zex 9086 . . . . . . . . . 10 ℤ ∈ V
3231mptex 5653 . . . . . . . . 9 (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
33 vex 2692 . . . . . . . . 9 𝑧 ∈ V
3432, 33fvex 5448 . . . . . . . 8 ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
3534ax-gen 1426 . . . . . . 7 𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
36 0z 9088 . . . . . . 7 0 ∈ ℤ
37 frecfnom 6305 . . . . . . 7 ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3835, 36, 37mp2an 423 . . . . . 6 frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
39 fndm 5229 . . . . . 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 2207 . . . 4 (ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ↔ ω ∈ ω)
4230, 41mtbir 661 . . 3 ¬ ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
43 fsnunfv 5628 . . 3 ((ω ∈ V ∧ +∞ ∈ ℝ* ∧ ¬ ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)) → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω) = +∞)
4410, 27, 42, 43mp3an 1316 . 2 ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω) = +∞
4526, 44eqtrdi 2189 1 (ω ≼ 𝐴 → (♯‘𝐴) = +∞)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1330   = wceq 1332   ∈ wcel 1481  {crab 2421  Vcvv 2689   ∪ cun 3073  ∅c0 3367  {csn 3531  ⟨cop 3534  ∪ cuni 3743   class class class wbr 3936   ↦ cmpt 3996  Ord word 4291  ωcom 4511  dom cdm 4546   ∘ ccom 4550  Rel wrel 4551  Fun wfun 5124   Fn wfn 5125  ‘cfv 5130  (class class class)co 5781  freccfrec 6294   ≼ cdom 6640  0cc0 7643  1c1 7644   + caddc 7646  +∞cpnf 7820  ℝ*cxr 7822  ℤcz 9077  ♯chash 10552 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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1re 7737  ax-addrcl 7740  ax-rnegex 7752 This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-recs 6209  df-frec 6295  df-dom 6643  df-pnf 7825  df-xr 7827  df-neg 7959  df-z 9078  df-ihash 10553 This theorem is referenced by:  filtinf  10569
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