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Theorem hashennn 10762
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 10758 . . . . 5 ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
21fveq1i 5518 . . . 4 (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))‘𝐴)
3 funmpt 5256 . . . . 5 Fun (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
4 hashennnuni 10761 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 𝑁𝐴) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = 𝑁)
54eqcomd 2183 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝑁 = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
6 nnfi 6874 . . . . . . . . . . 11 (𝑁 ∈ ω → 𝑁 ∈ Fin)
76adantr 276 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝑁 ∈ Fin)
8 simpr 110 . . . . . . . . . . 11 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝑁𝐴)
98ensymd 6785 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝐴𝑁)
10 enfii 6876 . . . . . . . . . 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 4009 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
1514adantr 276 . . . . . . . . . . . . 13 ((𝑥 = 𝐴𝑧 = 𝑁) → (𝑦𝑥𝑦𝐴))
1615rabbidv 2728 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑧 = 𝑁) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1716unieqd 3822 . . . . . . . . . . 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 4265 . . . . . . . . 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 4102 . . . . . . 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 4834 . . . . . . 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 5588 . . . . 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 2752 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝐴 ∈ V)
314, 12eqeltrd 2254 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑁𝐴) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ ω)
3214rabbidv 2728 . . . . . . . 8 (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
3332unieqd 3822 . . . . . . 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 5594 . . . . . 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 5521 . . 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 4608 . . . . . . 7 Ord ω
41 ordirr 4543 . . . . . . 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 5712 . . . 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 2739  cun 3129  {csn 3594  cop 3597   cuni 3811   class class class wbr 4005  {copab 4065  cmpt 4066  Ord word 4364  ωcom 4591  dom cdm 4628  ccom 4632  Fun wfun 5212  cfv 5218  (class class class)co 5877  freccfrec 6393  cen 6740  cdom 6741  Fincfn 6742  0cc0 7813  1c1 7814   + caddc 7816  +∞cpnf 7991  cz 9255  chash 10757
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 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
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 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-ihash 10758
This theorem is referenced by:  hashcl  10763  hashfz1  10765  hashen  10766  fihashdom  10785  hashun  10787
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