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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  *cxr 7822  cz 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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