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Theorem hashennn 10755
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 10751 . . . . 5 ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
21fveq1i 5516 . . . 4 (♯‘𝐴) = (((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))‘𝐴)
3 funmpt 5254 . . . . 5 Fun (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
4 hashennnuni 10754 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 𝑁𝐴) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} = 𝑁)
54eqcomd 2183 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝑁 = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
6 nnfi 6871 . . . . . . . . . . 11 (𝑁 ∈ ω → 𝑁 ∈ Fin)
76adantr 276 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝑁 ∈ Fin)
8 simpr 110 . . . . . . . . . . 11 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝑁𝐴)
98ensymd 6782 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝐴𝑁)
10 enfii 6873 . . . . . . . . . 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 4007 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
1514adantr 276 . . . . . . . . . . . . 13 ((𝑥 = 𝐴𝑧 = 𝑁) → (𝑦𝑥𝑦𝐴))
1615rabbidv 2726 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑧 = 𝑁) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
1716unieqd 3820 . . . . . . . . . . 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 4263 . . . . . . . . 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 4100 . . . . . . 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 4832 . . . . . . 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 5586 . . . . 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 2750 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑁𝐴) → 𝐴 ∈ V)
314, 12eqeltrd 2254 . . . . . 6 ((𝑁 ∈ ω ∧ 𝑁𝐴) → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴} ∈ ω)
3214rabbidv 2726 . . . . . . . 8 (𝑥 = 𝐴 → {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥} = {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝐴})
3332unieqd 3820 . . . . . . 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 5592 . . . . . 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 5519 . . 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 4606 . . . . . . 7 Ord ω
41 ordirr 4541 . . . . . . 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 5710 . . . 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 2737  cun 3127  {csn 3592  cop 3595   cuni 3809   class class class wbr 4003  {copab 4063  cmpt 4064  Ord word 4362  ωcom 4589  dom cdm 4626  ccom 4630  Fun wfun 5210  cfv 5216  (class class class)co 5874  freccfrec 6390  cen 6737  cdom 6738  Fincfn 6739  0cc0 7810  1c1 7811   + caddc 7813  +∞cpnf 7987  cz 9251  chash 10750
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 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-ihash 10751
This theorem is referenced by:  hashcl  10756  hashfz1  10758  hashen  10759  fihashdom  10778  hashun  10780
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