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Theorem r1om 10287
Description: The set of hereditarily finite sets is countable. See ackbij2 10286 for an explicit bijection that works without Infinity. See also r1omALT 10819. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Assertion
Ref Expression
r1om (𝑅1‘ω) ≈ ω

Proof of Theorem r1om
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omex 9686 . . . 4 ω ∈ V
2 limom 7892 . . . 4 Lim ω
3 r1lim 9815 . . . 4 ((ω ∈ V ∧ Lim ω) → (𝑅1‘ω) = 𝑎 ∈ ω (𝑅1𝑎))
41, 2, 3mp2an 690 . . 3 (𝑅1‘ω) = 𝑎 ∈ ω (𝑅1𝑎)
5 r1fnon 9810 . . . 4 𝑅1 Fn On
6 fnfun 6660 . . . 4 (𝑅1 Fn On → Fun 𝑅1)
7 funiunfv 7263 . . . 4 (Fun 𝑅1 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω))
85, 6, 7mp2b 10 . . 3 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω)
94, 8eqtri 2754 . 2 (𝑅1‘ω) = (𝑅1 “ ω)
10 iuneq1 5017 . . . . . . 7 (𝑒 = 𝑎 𝑓𝑒 ({𝑓} × 𝒫 𝑓) = 𝑓𝑎 ({𝑓} × 𝒫 𝑓))
11 sneq 4643 . . . . . . . . 9 (𝑓 = 𝑏 → {𝑓} = {𝑏})
12 pweq 4621 . . . . . . . . 9 (𝑓 = 𝑏 → 𝒫 𝑓 = 𝒫 𝑏)
1311, 12xpeq12d 5713 . . . . . . . 8 (𝑓 = 𝑏 → ({𝑓} × 𝒫 𝑓) = ({𝑏} × 𝒫 𝑏))
1413cbviunv 5048 . . . . . . 7 𝑓𝑎 ({𝑓} × 𝒫 𝑓) = 𝑏𝑎 ({𝑏} × 𝒫 𝑏)
1510, 14eqtrdi 2782 . . . . . 6 (𝑒 = 𝑎 𝑓𝑒 ({𝑓} × 𝒫 𝑓) = 𝑏𝑎 ({𝑏} × 𝒫 𝑏))
1615fveq2d 6905 . . . . 5 (𝑒 = 𝑎 → (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)) = (card‘ 𝑏𝑎 ({𝑏} × 𝒫 𝑏)))
1716cbvmptv 5266 . . . 4 (𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓))) = (𝑎 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑏𝑎 ({𝑏} × 𝒫 𝑏)))
18 dmeq 5910 . . . . . . . 8 (𝑐 = 𝑎 → dom 𝑐 = dom 𝑎)
1918pweqd 4624 . . . . . . 7 (𝑐 = 𝑎 → 𝒫 dom 𝑐 = 𝒫 dom 𝑎)
20 imaeq1 6064 . . . . . . . 8 (𝑐 = 𝑎 → (𝑐𝑑) = (𝑎𝑑))
2120fveq2d 6905 . . . . . . 7 (𝑐 = 𝑎 → ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐𝑑)) = ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑑)))
2219, 21mpteq12dv 5244 . . . . . 6 (𝑐 = 𝑎 → (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐𝑑))) = (𝑑 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑑))))
23 imaeq2 6065 . . . . . . . 8 (𝑑 = 𝑏 → (𝑎𝑑) = (𝑎𝑏))
2423fveq2d 6905 . . . . . . 7 (𝑑 = 𝑏 → ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑑)) = ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑏)))
2524cbvmptv 5266 . . . . . 6 (𝑑 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑑))) = (𝑏 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑏)))
2622, 25eqtrdi 2782 . . . . 5 (𝑐 = 𝑎 → (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐𝑑))) = (𝑏 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑏))))
2726cbvmptv 5266 . . . 4 (𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐𝑑)))) = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑏))))
28 eqid 2726 . . . 4 (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐𝑑)))), ∅) “ ω) = (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐𝑑)))), ∅) “ ω)
2917, 27, 28ackbij2 10286 . . 3 (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐𝑑)))), ∅) “ ω): (𝑅1 “ ω)–1-1-onto→ω
30 fvex 6914 . . . . 5 (𝑅1‘ω) ∈ V
319, 30eqeltrri 2823 . . . 4 (𝑅1 “ ω) ∈ V
3231f1oen 9004 . . 3 ( (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐𝑑)))), ∅) “ ω): (𝑅1 “ ω)–1-1-onto→ω → (𝑅1 “ ω) ≈ ω)
3329, 32ax-mp 5 . 2 (𝑅1 “ ω) ≈ ω
349, 33eqbrtri 5174 1 (𝑅1‘ω) ≈ ω
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  Vcvv 3462  cin 3946  c0 4325  𝒫 cpw 4607  {csn 4633   cuni 4913   ciun 5001   class class class wbr 5153  cmpt 5236   × cxp 5680  dom cdm 5682  cima 5685  Oncon0 6376  Lim wlim 6377  Fun wfun 6548   Fn wfn 6549  1-1-ontowf1o 6553  cfv 6554  ωcom 7876  reccrdg 8439  cen 8971  Fincfn 8974  𝑅1cr1 9805  cardccrd 9978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-oadd 8500  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-r1 9807  df-rank 9808  df-dju 9944  df-card 9982
This theorem is referenced by: (None)
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