Theorem ackbij3 46218

Description: The Ackermann bijection is a bijection. This is just a restatement of ackbij2 10344. (Contributed by Eric Schmidt, 24-Oct-2026.)

Assertion

Ref	Expression
ackbij3	⊢ ackBij: HF ⟶1-1ontoω

Proof of Theorem ackbij3

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq1 4971	. . . . 5 ⊢ (𝑧 = 𝑢 → ∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤) = ∪ 𝑤 ∈ 𝑢 ({𝑤} × 𝒫 𝑤))
2	1	fveq2d 6909	. . . 4 ⊢ (𝑧 = 𝑢 → (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑤 ∈ 𝑢 ({𝑤} × 𝒫 𝑤)))
3	2	cbvmptv 5212	. . 3 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑢 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑢 ({𝑤} × 𝒫 𝑤)))
4		sneq 4594	. . . . . . 7 ⊢ (𝑤 = 𝑣 → {𝑤} = {𝑣})
5		pweq 4574	. . . . . . 7 ⊢ (𝑤 = 𝑣 → 𝒫 𝑤 = 𝒫 𝑣)
6	4, 5	xpeq12d 5686	. . . . . 6 ⊢ (𝑤 = 𝑣 → ({𝑤} × 𝒫 𝑤) = ({𝑣} × 𝒫 𝑣))
7	6	cbviunv 5000	. . . . 5 ⊢ ∪ 𝑤 ∈ 𝑢 ({𝑤} × 𝒫 𝑤) = ∪ 𝑣 ∈ 𝑢 ({𝑣} × 𝒫 𝑣)
8	7	fveq2i 6908	. . . 4 ⊢ (card‘∪ 𝑤 ∈ 𝑢 ({𝑤} × 𝒫 𝑤)) = (card‘∪ 𝑣 ∈ 𝑢 ({𝑣} × 𝒫 𝑣))
9	8	mpteq2i 5204	. . 3 ⊢ (𝑢 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑢 ({𝑤} × 𝒫 𝑤))) = (𝑢 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑣 ∈ 𝑢 ({𝑣} × 𝒫 𝑣)))
10	3, 9	eqtri 2786	. 2 ⊢ (𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤))) = (𝑢 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑣 ∈ 𝑢 ({𝑣} × 𝒫 𝑣)))
11		dmeq 5891	. . . . . 6 ⊢ (𝑥 = 𝑢 → dom 𝑥 = dom 𝑢)
12	11	pweqd 4577	. . . . 5 ⊢ (𝑥 = 𝑢 → 𝒫 dom 𝑥 = 𝒫 dom 𝑢)
13		imaeq1 6058	. . . . . 6 ⊢ (𝑥 = 𝑢 → (𝑥 “ 𝑦) = (𝑢 “ 𝑦))
14	13	fveq2d 6909	. . . . 5 ⊢ (𝑥 = 𝑢 → ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑥 “ 𝑦)) = ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑢 “ 𝑦)))
15	12, 14	mpteq12dv 5195	. . . 4 ⊢ (𝑥 = 𝑢 → (𝑦 ∈ 𝒫 dom 𝑥 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑥 “ 𝑦))) = (𝑦 ∈ 𝒫 dom 𝑢 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑢 “ 𝑦))))
16	15	cbvmptv 5212	. . 3 ⊢ (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑥 “ 𝑦)))) = (𝑢 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑢 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑢 “ 𝑦))))
17		imaeq2 6059	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑢 “ 𝑦) = (𝑢 “ 𝑣))
18	17	fveq2d 6909	. . . . 5 ⊢ (𝑦 = 𝑣 → ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑢 “ 𝑦)) = ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑢 “ 𝑣)))
19	18	cbvmptv 5212	. . . 4 ⊢ (𝑦 ∈ 𝒫 dom 𝑢 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑢 “ 𝑦))) = (𝑣 ∈ 𝒫 dom 𝑢 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑢 “ 𝑣)))
20	19	mpteq2i 5204	. . 3 ⊢ (𝑢 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑢 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑢 “ 𝑦)))) = (𝑢 ∈ V ↦ (𝑣 ∈ 𝒫 dom 𝑢 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑢 “ 𝑣))))
21	16, 20	eqtri 2786	. 2 ⊢ (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑥 “ 𝑦)))) = (𝑢 ∈ V ↦ (𝑣 ∈ 𝒫 dom 𝑢 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑢 “ 𝑣))))
22		df-ackbij 46217	. 2 ⊢ ackBij = ∪ (rec((𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑥 “ 𝑦)))), ∅) “ ω)
23	10, 21, 22	ackbij2 10344	1 ⊢ ackBij: HF ⟶1-1ontoω

This proof depends on syntax axioms:  Vcvv 3454   ∩ cin 4092  𝒫 cpw 4560  {csn 4588  ∪ ciun 4954   ↦ cmpt 5189   × cxp 5653  dom cdm 5655   “ cima 5658  ⟶1-1ontowf1o 6554  ‘cfv 6555  ωcom 7898  Fincfn 8998   HF chf 9928  cardccrd 10040  ackBijcackbij 46216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-r1 9792  df-rank 9793  df-hf 9929  df-dju 10006  df-card 10044  df-ackbij 46217

This theorem is used by:  hfstructwe  46232