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Theorem hashennn 10760
Description: The size of a set equinumerous to an element of ω. (Contributed by Jim Kingdon, 21-Feb-2022.)
Assertion
Ref Expression
hashennn ((𝑁 ∈ ω ∧ 𝑁𝐴) → (♯‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑁

Proof of Theorem hashennn
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ihash 10756 . . . . 5 ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
21fveq1i 5517 . . . 4 (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))‘𝐴)
3 funmpt 5255 . . . . 5 Fun (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
4 hashennnuni 10759 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 𝑁𝐴) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = 𝑁)
54eqcomd 2183 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝑁 = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
6 nnfi 6872 . . . . . . . . . . 11 (𝑁 ∈ ω → 𝑁 ∈ Fin)
76adantr 276 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝑁 ∈ Fin)
8 simpr 110 . . . . . . . . . . 11 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝑁𝐴)
98ensymd 6783 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝐴𝑁)
10 enfii 6874 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝐴𝑁) → 𝐴 ∈ Fin)
117, 9, 10syl2anc 411 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝐴 ∈ Fin)
12 simpl 109 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝑁 ∈ ω)
13 simpr 110 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑧 = 𝑁) → 𝑧 = 𝑁)
14 breq2 4008 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
1514adantr 276 . . . . . . . . . . . . 13 ((𝑥 = 𝐴𝑧 = 𝑁) → (𝑦𝑥𝑦𝐴))
1615rabbidv 2727 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑧 = 𝑁) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1716unieqd 3821 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑧 = 𝑁) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1813, 17eqeq12d 2192 . . . . . . . . . 10 ((𝑥 = 𝐴𝑧 = 𝑁) → (𝑧 = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} ↔ 𝑁 = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}))
1918opelopabga 4264 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ 𝑁 ∈ ω) → (⟨𝐴, 𝑁⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}} ↔ 𝑁 = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}))
2011, 12, 19syl2anc 411 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝑁𝐴) → (⟨𝐴, 𝑁⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}} ↔ 𝑁 = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴}))
215, 20mpbird 167 . . . . . . 7 ((𝑁 ∈ ω ∧ 𝑁𝐴) → ⟨𝐴, 𝑁⟩ ∈ {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}})
22 mptv 4101 . . . . . . 7 (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}}
2321, 22eleqtrrdi 2271 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑁𝐴) → ⟨𝐴, 𝑁⟩ ∈ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
24 opeldmg 4833 . . . . . . 7 ((𝐴 ∈ Fin ∧ 𝑁 ∈ ω) → (⟨𝐴, 𝑁⟩ ∈ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) → 𝐴 ∈ dom (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})))
2511, 12, 24syl2anc 411 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑁𝐴) → (⟨𝐴, 𝑁⟩ ∈ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) → 𝐴 ∈ dom (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})))
2623, 25mpd 13 . . . . 5 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝐴 ∈ dom (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
27 fvco 5587 . . . . 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)))
283, 26, 27sylancr 414 . . . 4 ((𝑁 ∈ ω ∧ 𝑁𝐴) → (((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)))
292, 28eqtrid 2222 . . 3 ((𝑁 ∈ ω ∧ 𝑁𝐴) → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴)))
3011elexd 2751 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝐴 ∈ V)
314, 12eqeltrd 2254 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑁𝐴) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ ω)
3214rabbidv 2727 . . . . . . . 8 (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
3332unieqd 3821 . . . . . . 7 (𝑥 = 𝐴 {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
34 eqid 2177 . . . . . . 7 (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}) = (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
3533, 34fvmptg 5593 . . . . . 6 ((𝐴 ∈ V ∧ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ ω) → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
3630, 31, 35syl2anc 411 . . . . 5 ((𝑁 ∈ ω ∧ 𝑁𝐴) → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
3736, 4eqtrd 2210 . . . 4 ((𝑁 ∈ ω ∧ 𝑁𝐴) → ((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴) = 𝑁)
3837fveq2d 5520 . . 3 ((𝑁 ∈ ω ∧ 𝑁𝐴) → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘((𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})‘𝐴)) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁))
3929, 38eqtrd 2210 . 2 ((𝑁 ∈ ω ∧ 𝑁𝐴) → (♯‘𝐴) = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁))
40 ordom 4607 . . . . . . 7 Ord ω
41 ordirr 4542 . . . . . . 7 (Ord ω → ¬ ω ∈ ω)
4240, 41ax-mp 5 . . . . . 6 ¬ ω ∈ ω
43 eleq1 2240 . . . . . 6 (ω = 𝑁 → (ω ∈ ω ↔ 𝑁 ∈ ω))
4442, 43mtbii 674 . . . . 5 (ω = 𝑁 → ¬ 𝑁 ∈ ω)
4544necon2ai 2401 . . . 4 (𝑁 ∈ ω → ω ≠ 𝑁)
46 fvunsng 5711 . . . 4 ((𝑁 ∈ ω ∧ ω ≠ 𝑁) → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))
4745, 46mpdan 421 . . 3 (𝑁 ∈ ω → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))
4847adantr 276 . 2 ((𝑁 ∈ ω ∧ 𝑁𝐴) → ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})‘𝑁) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))
4939, 48eqtrd 2210 1 ((𝑁 ∈ ω ∧ 𝑁𝐴) → (♯‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wne 2347  {crab 2459  Vcvv 2738  cun 3128  {csn 3593  cop 3596   cuni 3810   class class class wbr 4004  {copab 4064  cmpt 4065  Ord word 4363  ωcom 4590  dom cdm 4627  ccom 4631  Fun wfun 5211  cfv 5217  (class class class)co 5875  freccfrec 6391  cen 6738  cdom 6739  Fincfn 6740  0cc0 7811  1c1 7812   + caddc 7814  +∞cpnf 7989  cz 9253  chash 10755
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-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  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-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-er 6535  df-en 6741  df-dom 6742  df-fin 6743  df-ihash 10756
This theorem is referenced by:  hashcl  10761  hashfz1  10763  hashen  10764  fihashdom  10783  hashun  10785
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