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Theorem r1om 9319
Description: The set of hereditarily finite sets is countable. See ackbij2 9318 for an explicit bijection that works without Infinity. See also r1omALT 9851. (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 8755 . . . 4 ω ∈ V
2 limom 7278 . . . 4 Lim ω
3 r1lim 8850 . . . 4 ((ω ∈ V ∧ Lim ω) → (𝑅1‘ω) = 𝑎 ∈ ω (𝑅1𝑎))
41, 2, 3mp2an 683 . . 3 (𝑅1‘ω) = 𝑎 ∈ ω (𝑅1𝑎)
5 r1fnon 8845 . . . 4 𝑅1 Fn On
6 fnfun 6166 . . . 4 (𝑅1 Fn On → Fun 𝑅1)
7 funiunfv 6698 . . . 4 (Fun 𝑅1 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω))
85, 6, 7mp2b 10 . . 3 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω)
94, 8eqtri 2787 . 2 (𝑅1‘ω) = (𝑅1 “ ω)
10 iuneq1 4690 . . . . . . 7 (𝑒 = 𝑎 𝑓𝑒 ({𝑓} × 𝒫 𝑓) = 𝑓𝑎 ({𝑓} × 𝒫 𝑓))
11 sneq 4344 . . . . . . . . 9 (𝑓 = 𝑏 → {𝑓} = {𝑏})
12 pweq 4318 . . . . . . . . 9 (𝑓 = 𝑏 → 𝒫 𝑓 = 𝒫 𝑏)
1311, 12xpeq12d 5308 . . . . . . . 8 (𝑓 = 𝑏 → ({𝑓} × 𝒫 𝑓) = ({𝑏} × 𝒫 𝑏))
1413cbviunv 4715 . . . . . . 7 𝑓𝑎 ({𝑓} × 𝒫 𝑓) = 𝑏𝑎 ({𝑏} × 𝒫 𝑏)
1510, 14syl6eq 2815 . . . . . 6 (𝑒 = 𝑎 𝑓𝑒 ({𝑓} × 𝒫 𝑓) = 𝑏𝑎 ({𝑏} × 𝒫 𝑏))
1615fveq2d 6379 . . . . 5 (𝑒 = 𝑎 → (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)) = (card‘ 𝑏𝑎 ({𝑏} × 𝒫 𝑏)))
1716cbvmptv 4909 . . . 4 (𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓))) = (𝑎 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑏𝑎 ({𝑏} × 𝒫 𝑏)))
18 dmeq 5492 . . . . . . . 8 (𝑐 = 𝑎 → dom 𝑐 = dom 𝑎)
1918pweqd 4320 . . . . . . 7 (𝑐 = 𝑎 → 𝒫 dom 𝑐 = 𝒫 dom 𝑎)
20 imaeq1 5643 . . . . . . . 8 (𝑐 = 𝑎 → (𝑐𝑑) = (𝑎𝑑))
2120fveq2d 6379 . . . . . . 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 4892 . . . . . 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 5644 . . . . . . . 8 (𝑑 = 𝑏 → (𝑎𝑑) = (𝑎𝑏))
2423fveq2d 6379 . . . . . . 7 (𝑑 = 𝑏 → ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑑)) = ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑓𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎𝑏)))
2524cbvmptv 4909 . . . . . 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 2815 . . . . 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 4909 . . . 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 2765 . . . 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 9318 . . 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 6388 . . . . 5 (𝑅1‘ω) ∈ V
319, 30eqeltrri 2841 . . . 4 (𝑅1 “ ω) ∈ V
3231f1oen 8181 . . 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 4830 1 (𝑅1‘ω) ≈ ω
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  wcel 2155  Vcvv 3350  cin 3731  c0 4079  𝒫 cpw 4315  {csn 4334   cuni 4594   ciun 4676   class class class wbr 4809  cmpt 4888   × cxp 5275  dom cdm 5277  cima 5280  Oncon0 5908  Lim wlim 5909  Fun wfun 6062   Fn wfn 6063  1-1-ontowf1o 6067  cfv 6068  ωcom 7263  reccrdg 7709  cen 8157  Fincfn 8160  𝑅1cr1 8840  cardccrd 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-r1 8842  df-rank 8843  df-card 9016  df-cda 9243
This theorem is referenced by: (None)
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