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Theorem hashinfom 10301
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 10299 . . . . 5 ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
21fveq1i 5341 . . . 4 (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))‘𝐴)
3 funmpt 5086 . . . . 5 Fun (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
4 funrel 5066 . . . . . . 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 4437 . . . . . . 7 ∅ ∈ ω
7 reldom 6542 . . . . . . . . . 10 Rel ≼
87brrelex2i 4511 . . . . . . . . 9 (ω ≼ 𝐴𝐴 ∈ V)
9 hashinfuni 10300 . . . . . . . . . 10 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = ω)
10 omex 4436 . . . . . . . . . 10 ω ∈ V
119, 10syl6eqel 2185 . . . . . . . . 9 (ω ≼ 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ V)
12 breq2 3871 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
1312rabbidv 2622 . . . . . . . . . . 11 (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1413unieqd 3686 . . . . . . . . . 10 (𝑥 = 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
15 eqid 2095 . . . . . . . . . 10 (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) = (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
1614, 15fvmptg 5415 . . . . . . . . 9 ((𝐴 ∈ V ∧ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ V) → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
178, 11, 16syl2anc 404 . . . . . . . 8 (ω ≼ 𝐴 → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1817, 9eqtrd 2127 . . . . . . 7 (ω ≼ 𝐴 → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = ω)
196, 18syl5eleqr 2184 . . . . . 6 (ω ≼ 𝐴 → ∅ ∈ ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴))
20 relelfvdm 5371 . . . . . 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 406 . . . . 5 (ω ≼ 𝐴𝐴 ∈ dom (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
22 fvco 5409 . . . . 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 406 . . . 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 2139 . . 3 (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴)))
2518fveq2d 5344 . . 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 2127 . 2 (ω ≼ 𝐴 → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω))
27 pnfxr 7637 . . 3 +∞ ∈ ℝ*
28 ordom 4449 . . . . 5 Ord ω
29 ordirr 4386 . . . . 5 (Ord ω → ¬ ω ∈ ω)
3028, 29ax-mp 7 . . . 4 ¬ ω ∈ ω
31 zex 8857 . . . . . . . . . 10 ℤ ∈ V
3231mptex 5562 . . . . . . . . 9 (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
33 vex 2636 . . . . . . . . 9 𝑧 ∈ V
3432, 33fvex 5360 . . . . . . . 8 ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
3534ax-gen 1390 . . . . . . 7 𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
36 0z 8859 . . . . . . 7 0 ∈ ℤ
37 frecfnom 6204 . . . . . . 7 ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3835, 36, 37mp2an 418 . . . . . 6 frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
39 fndm 5147 . . . . . 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 2161 . . . 4 (ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ↔ ω ∈ ω)
4230, 41mtbir 634 . . 3 ¬ ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
43 fsnunfv 5537 . . 3 ((ω ∈ V ∧ +∞ ∈ ℝ* ∧ ¬ ω ∈ dom frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)) → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω) = +∞)
4410, 27, 42, 43mp3an 1280 . 2 ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘ω) = +∞
4526, 44syl6eq 2143 1 (ω ≼ 𝐴 → (♯‘𝐴) = +∞)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1294   = wceq 1296  wcel 1445  {crab 2374  Vcvv 2633  cun 3011  c0 3302  {csn 3466  cop 3469   cuni 3675   class class class wbr 3867  cmpt 3921  Ord word 4213  ωcom 4433  dom cdm 4467  ccom 4471  Rel wrel 4472  Fun wfun 5043   Fn wfn 5044  cfv 5049  (class class class)co 5690  freccfrec 6193  cdom 6536  0cc0 7447  1c1 7448   + caddc 7450  +∞cpnf 7616  *cxr 7618  cz 8848  chash 10298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1re 7536  ax-addrcl 7539  ax-rnegex 7551
This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-recs 6108  df-frec 6194  df-dom 6539  df-pnf 7621  df-xr 7623  df-neg 7753  df-z 8849  df-ihash 10299
This theorem is referenced by:  filtinf  10315
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