Theorem hfom 10322

Description: The set of hereditarily finite sets is countable. See ackbij2 10321 for an explicit bijection that works without Infinity. See also hfomALT 10862. (Contributed by Stefan O'Rear, 18-Nov-2014.) Restate using the defined HF symbol. (Revised by Eric Schmidt, 24-Sep-2026.)

Assertion

Ref	Expression
hfom	⊢ HF ≈ ω

Proof of Theorem hfom

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq1 4967	. . . . . 6 ⊢ (𝑒 = 𝑎 → ∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓) = ∪ 𝑓 ∈ 𝑎 ({𝑓} × 𝒫 𝑓))
2		sneq 4593	. . . . . . . 8 ⊢ (𝑓 = 𝑏 → {𝑓} = {𝑏})
3		pweq 4570	. . . . . . . 8 ⊢ (𝑓 = 𝑏 → 𝒫 𝑓 = 𝒫 𝑏)
4	2, 3	xpeq12d 5681	. . . . . . 7 ⊢ (𝑓 = 𝑏 → ({𝑓} × 𝒫 𝑓) = ({𝑏} × 𝒫 𝑏))
5	4	cbviunv 4996	. . . . . 6 ⊢ ∪ 𝑓 ∈ 𝑎 ({𝑓} × 𝒫 𝑓) = ∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)
6	1, 5	eqtrdi 2811	. . . . 5 ⊢ (𝑒 = 𝑎 → ∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓) = ∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏))
7	6	fveq2d 6889	. . . 4 ⊢ (𝑒 = 𝑎 → (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)) = (card‘∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)))
8	7	cbvmptv 5208	. . 3 ⊢ (𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓))) = (𝑎 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)))
9		dmeq 5884	. . . . . . 7 ⊢ (𝑐 = 𝑎 → dom 𝑐 = dom 𝑎)
10	9	pweqd 4573	. . . . . 6 ⊢ (𝑐 = 𝑎 → 𝒫 dom 𝑐 = 𝒫 dom 𝑎)
11		imaeq1 6046	. . . . . . 7 ⊢ (𝑐 = 𝑎 → (𝑐 “ 𝑑) = (𝑎 “ 𝑑))
12	11	fveq2d 6889	. . . . . 6 ⊢ (𝑐 = 𝑎 → ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)) = ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑑)))
13	10, 12	mpteq12dv 5191	. . . . 5 ⊢ (𝑐 = 𝑎 → (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑))) = (𝑑 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑑))))
14		imaeq2 6047	. . . . . . 7 ⊢ (𝑑 = 𝑏 → (𝑎 “ 𝑑) = (𝑎 “ 𝑏))
15	14	fveq2d 6889	. . . . . 6 ⊢ (𝑑 = 𝑏 → ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑑)) = ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑏)))
16	15	cbvmptv 5208	. . . . 5 ⊢ (𝑑 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑑))) = (𝑏 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑏)))
17	13, 16	eqtrdi 2811	. . . 4 ⊢ (𝑐 = 𝑎 → (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑))) = (𝑏 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑏))))
18	17	cbvmptv 5208	. . 3 ⊢ (𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)))) = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑏))))
19		eqid 2760	. . 3 ⊢ ∪ (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)))), ∅) “ ω) = ∪ (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)))), ∅) “ ω)
20	8, 18, 19	ackbij2 10321	. 2 ⊢ ∪ (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)))), ∅) “ ω): HF –1-1-onto→ω
21		dfhf2 9907	. . . 4 ⊢ HF = (𝑅1‘ω)
22	21	fvexi 6899	. . 3 ⊢ HF ∈ V
23	22	f1oen 9000	. 2 ⊢ (∪ (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)))), ∅) “ ω): HF –1-1-onto→ω → HF ≈ ω)
24	20, 23	ax-mp 5	1 ⊢ HF ≈ ω

This proof depends on syntax axioms:  Vcvv 3450   ∩ cin 4089  ∅c0 4278  𝒫 cpw 4556  {csn 4583  ∪ cuni 4866  ∪ ciun 4950   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648  dom cdm 5650   “ cima 5653  –1-1-onto→wf1o 6537  ‘cfv 6538  ωcom 7877  reccrdg 8418   ≈ cen 8971  Fincfn 8974  𝑅₁cr1 9767   HF chf 9905  cardccrd 10017

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9643

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-2o 8478  df-oadd 8481  df-er 8718  df-map 8850  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-r1 9769  df-rank 9770  df-hf 9906  df-dju 9983  df-card 10021

This theorem is used by: (None)