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Theorem r1om 10281
Description: The set of hereditarily finite sets is countable. See ackbij2 10280 for an explicit bijection that works without Infinity. See also r1omALT 10814. (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 9681 . . . 4 ω ∈ V
2 limom 7903 . . . 4 Lim ω
3 r1lim 9810 . . . 4 ((ω ∈ V ∧ Lim ω) → (𝑅1‘ω) = 𝑎 ∈ ω (𝑅1𝑎))
41, 2, 3mp2an 692 . . 3 (𝑅1‘ω) = 𝑎 ∈ ω (𝑅1𝑎)
5 r1fnon 9805 . . . 4 𝑅1 Fn On
6 fnfun 6669 . . . 4 (𝑅1 Fn On → Fun 𝑅1)
7 funiunfv 7268 . . . 4 (Fun 𝑅1 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω))
85, 6, 7mp2b 10 . . 3 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω)
94, 8eqtri 2763 . 2 (𝑅1‘ω) = (𝑅1 “ ω)
10 iuneq1 5013 . . . . . . 7 (𝑒 = 𝑎 𝑓𝑒 ({𝑓} × 𝒫 𝑓) = 𝑓𝑎 ({𝑓} × 𝒫 𝑓))
11 sneq 4641 . . . . . . . . 9 (𝑓 = 𝑏 → {𝑓} = {𝑏})
12 pweq 4619 . . . . . . . . 9 (𝑓 = 𝑏 → 𝒫 𝑓 = 𝒫 𝑏)
1311, 12xpeq12d 5720 . . . . . . . 8 (𝑓 = 𝑏 → ({𝑓} × 𝒫 𝑓) = ({𝑏} × 𝒫 𝑏))
1413cbviunv 5045 . . . . . . 7 𝑓𝑎 ({𝑓} × 𝒫 𝑓) = 𝑏𝑎 ({𝑏} × 𝒫 𝑏)
1510, 14eqtrdi 2791 . . . . . 6 (𝑒 = 𝑎 𝑓𝑒 ({𝑓} × 𝒫 𝑓) = 𝑏𝑎 ({𝑏} × 𝒫 𝑏))
1615fveq2d 6911 . . . . 5 (𝑒 = 𝑎 → (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)) = (card‘ 𝑏𝑎 ({𝑏} × 𝒫 𝑏)))
1716cbvmptv 5261 . . . 4 (𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓))) = (𝑎 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑏𝑎 ({𝑏} × 𝒫 𝑏)))
18 dmeq 5917 . . . . . . . 8 (𝑐 = 𝑎 → dom 𝑐 = dom 𝑎)
1918pweqd 4622 . . . . . . 7 (𝑐 = 𝑎 → 𝒫 dom 𝑐 = 𝒫 dom 𝑎)
20 imaeq1 6075 . . . . . . . 8 (𝑐 = 𝑎 → (𝑐𝑑) = (𝑎𝑑))
2120fveq2d 6911 . . . . . . 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 5239 . . . . . 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 6076 . . . . . . . 8 (𝑑 = 𝑏 → (𝑎𝑑) = (𝑎𝑏))
2423fveq2d 6911 . . . . . . 7 (𝑑 = 𝑏 → ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑑)) = ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑏)))
2524cbvmptv 5261 . . . . . 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 2791 . . . . 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 5261 . . . 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 2735 . . . 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 10280 . . 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 6920 . . . . 5 (𝑅1‘ω) ∈ V
319, 30eqeltrri 2836 . . . 4 (𝑅1 “ ω) ∈ V
3231f1oen 9012 . . 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 5169 1 (𝑅1‘ω) ≈ ω
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912   ciun 4996   class class class wbr 5148  cmpt 5231   × cxp 5687  dom cdm 5689  cima 5692  Oncon0 6386  Lim wlim 6387  Fun wfun 6557   Fn wfn 6558  1-1-ontowf1o 6562  cfv 6563  ωcom 7887  reccrdg 8448  cen 8981  Fincfn 8984  𝑅1cr1 9800  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-r1 9802  df-rank 9803  df-dju 9939  df-card 9977
This theorem is referenced by: (None)
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