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Theorem r1om 9104
Description: The set of hereditarily finite sets is countable. See ackbij2 9103 for an explicit bijection that works without Infinity. See also r1omALT 9636. (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 8578 . . . 4 ω ∈ V
2 limom 7122 . . . 4 Lim ω
3 r1lim 8673 . . . 4 ((ω ∈ V ∧ Lim ω) → (𝑅1‘ω) = 𝑎 ∈ ω (𝑅1𝑎))
41, 2, 3mp2an 708 . . 3 (𝑅1‘ω) = 𝑎 ∈ ω (𝑅1𝑎)
5 r1fnon 8668 . . . 4 𝑅1 Fn On
6 fnfun 6026 . . . 4 (𝑅1 Fn On → Fun 𝑅1)
7 funiunfv 6546 . . . 4 (Fun 𝑅1 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω))
85, 6, 7mp2b 10 . . 3 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω)
94, 8eqtri 2673 . 2 (𝑅1‘ω) = (𝑅1 “ ω)
10 iuneq1 4566 . . . . . . 7 (𝑒 = 𝑎 𝑓𝑒 ({𝑓} × 𝒫 𝑓) = 𝑓𝑎 ({𝑓} × 𝒫 𝑓))
11 sneq 4220 . . . . . . . . 9 (𝑓 = 𝑏 → {𝑓} = {𝑏})
12 pweq 4194 . . . . . . . . 9 (𝑓 = 𝑏 → 𝒫 𝑓 = 𝒫 𝑏)
1311, 12xpeq12d 5174 . . . . . . . 8 (𝑓 = 𝑏 → ({𝑓} × 𝒫 𝑓) = ({𝑏} × 𝒫 𝑏))
1413cbviunv 4591 . . . . . . 7 𝑓𝑎 ({𝑓} × 𝒫 𝑓) = 𝑏𝑎 ({𝑏} × 𝒫 𝑏)
1510, 14syl6eq 2701 . . . . . 6 (𝑒 = 𝑎 𝑓𝑒 ({𝑓} × 𝒫 𝑓) = 𝑏𝑎 ({𝑏} × 𝒫 𝑏))
1615fveq2d 6233 . . . . 5 (𝑒 = 𝑎 → (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)) = (card‘ 𝑏𝑎 ({𝑏} × 𝒫 𝑏)))
1716cbvmptv 4783 . . . 4 (𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓))) = (𝑎 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑏𝑎 ({𝑏} × 𝒫 𝑏)))
18 dmeq 5356 . . . . . . . 8 (𝑐 = 𝑎 → dom 𝑐 = dom 𝑎)
1918pweqd 4196 . . . . . . 7 (𝑐 = 𝑎 → 𝒫 dom 𝑐 = 𝒫 dom 𝑎)
20 imaeq1 5496 . . . . . . . 8 (𝑐 = 𝑎 → (𝑐𝑑) = (𝑎𝑑))
2120fveq2d 6233 . . . . . . 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 4766 . . . . . 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 5497 . . . . . . . 8 (𝑑 = 𝑏 → (𝑎𝑑) = (𝑎𝑏))
2423fveq2d 6233 . . . . . . 7 (𝑑 = 𝑏 → ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑑)) = ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑏)))
2524cbvmptv 4783 . . . . . 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, 25syl6eq 2701 . . . . 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 4783 . . . 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 2651 . . . 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 9103 . . 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 6239 . . . . 5 (𝑅1‘ω) ∈ V
319, 30eqeltrri 2727 . . . 4 (𝑅1 “ ω) ∈ V
3231f1oen 8018 . . 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 4706 1 (𝑅1‘ω) ≈ ω
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  cima 5146  Oncon0 5761  Lim wlim 5762  Fun wfun 5920   Fn wfn 5921  1-1-ontowf1o 5925  cfv 5926  ωcom 7107  reccrdg 7550  cen 7994  Fincfn 7997  𝑅1cr1 8663  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-r1 8665  df-rank 8666  df-card 8803  df-cda 9028
This theorem is referenced by: (None)
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