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Theorem r1om 9026
Description: The set of hereditarily finite sets is countable. See ackbij2 9025 for an explicit bijection that works without Infinity. See also r1omALT 9558. (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 8500 . . . 4 ω ∈ V
2 limom 7042 . . . 4 Lim ω
3 r1lim 8595 . . . 4 ((ω ∈ V ∧ Lim ω) → (𝑅1‘ω) = 𝑎 ∈ ω (𝑅1𝑎))
41, 2, 3mp2an 707 . . 3 (𝑅1‘ω) = 𝑎 ∈ ω (𝑅1𝑎)
5 r1fnon 8590 . . . 4 𝑅1 Fn On
6 fnfun 5956 . . . 4 (𝑅1 Fn On → Fun 𝑅1)
7 funiunfv 6471 . . . 4 (Fun 𝑅1 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω))
85, 6, 7mp2b 10 . . 3 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω)
94, 8eqtri 2643 . 2 (𝑅1‘ω) = (𝑅1 “ ω)
10 iuneq1 4507 . . . . . . 7 (𝑒 = 𝑎 𝑓𝑒 ({𝑓} × 𝒫 𝑓) = 𝑓𝑎 ({𝑓} × 𝒫 𝑓))
11 sneq 4165 . . . . . . . . 9 (𝑓 = 𝑏 → {𝑓} = {𝑏})
12 pweq 4139 . . . . . . . . 9 (𝑓 = 𝑏 → 𝒫 𝑓 = 𝒫 𝑏)
1311, 12xpeq12d 5110 . . . . . . . 8 (𝑓 = 𝑏 → ({𝑓} × 𝒫 𝑓) = ({𝑏} × 𝒫 𝑏))
1413cbviunv 4532 . . . . . . 7 𝑓𝑎 ({𝑓} × 𝒫 𝑓) = 𝑏𝑎 ({𝑏} × 𝒫 𝑏)
1510, 14syl6eq 2671 . . . . . 6 (𝑒 = 𝑎 𝑓𝑒 ({𝑓} × 𝒫 𝑓) = 𝑏𝑎 ({𝑏} × 𝒫 𝑏))
1615fveq2d 6162 . . . . 5 (𝑒 = 𝑎 → (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)) = (card‘ 𝑏𝑎 ({𝑏} × 𝒫 𝑏)))
1716cbvmptv 4720 . . . 4 (𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓))) = (𝑎 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑏𝑎 ({𝑏} × 𝒫 𝑏)))
18 dmeq 5294 . . . . . . . 8 (𝑐 = 𝑎 → dom 𝑐 = dom 𝑎)
1918pweqd 4141 . . . . . . 7 (𝑐 = 𝑎 → 𝒫 dom 𝑐 = 𝒫 dom 𝑎)
20 imaeq1 5430 . . . . . . . 8 (𝑐 = 𝑎 → (𝑐𝑑) = (𝑎𝑑))
2120fveq2d 6162 . . . . . . 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 4703 . . . . . 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 5431 . . . . . . . 8 (𝑑 = 𝑏 → (𝑎𝑑) = (𝑎𝑏))
2423fveq2d 6162 . . . . . . 7 (𝑑 = 𝑏 → ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑑)) = ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑏)))
2524cbvmptv 4720 . . . . . 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 2671 . . . . 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 4720 . . . 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 2621 . . . 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 9025 . . 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 6168 . . . . 5 (𝑅1‘ω) ∈ V
319, 30eqeltrri 2695 . . . 4 (𝑅1 “ ω) ∈ V
3231f1oen 7936 . . 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 4644 1 (𝑅1‘ω) ≈ ω
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3190  cin 3559  c0 3897  𝒫 cpw 4136  {csn 4155   cuni 4409   ciun 4492   class class class wbr 4623  cmpt 4683   × cxp 5082  dom cdm 5084  cima 5087  Oncon0 5692  Lim wlim 5693  Fun wfun 5851   Fn wfn 5852  1-1-ontowf1o 5856  cfv 5857  ωcom 7027  reccrdg 7465  cen 7912  Fincfn 7915  𝑅1cr1 8585  cardccrd 8721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-r1 8587  df-rank 8588  df-card 8725  df-cda 8950
This theorem is referenced by: (None)
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