Theorem jf1o 35962

Description: 𝐽 is a bijection from ((On × On) × 9_o) to On. Proposition 15.3 of [TakeutiZaring] p. 155. (Contributed by BTernaryTau, 6-Oct-2026.)

Assertion

Ref	Expression
jf1o	⊢ 𝐽:((On × On) × 9o)–1-1-onto→On

Proof of Theorem jf1o

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

Step	Hyp	Ref	Expression
1		jf1 35961	. 2 ⊢ 𝐽:((On × On) × 9o)–1-1→On
2		f1f 6783	. . . 4 ⊢ (𝐽:((On × On) × 9o)–1-1→On → 𝐽:((On × On) × 9o)⟶On)
3		frn 6722	. . . 4 ⊢ (𝐽:((On × On) × 9o)⟶On → ran 𝐽 ⊆ On)
4	1, 2, 3	mp2b 10	. . 3 ⊢ ran 𝐽 ⊆ On
5		ssab 3987	. . . . 5 ⊢ (On ⊆ {𝑤 ∣ ∃𝑧 ∈ ((On × On) × 9o)𝑤 = ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧))} ↔ ∀𝑤(𝑤 ∈ On → ∃𝑧 ∈ ((On × On) × 9o)𝑤 = ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧))))
6		9on 35843	. . . . . . . 8 ⊢ 9o ∈ On
7		df-9o 35838	. . . . . . . . 9 ⊢ 9o = suc 8o
8		nsuceq0 6451	. . . . . . . . 9 ⊢ suc 8o ≠ ∅
9	7, 8	eqnetri 3025	. . . . . . . 8 ⊢ 9o ≠ ∅
10		omeulem1 8598	. . . . . . . 8 ⊢ ((9o ∈ On ∧ 𝑤 ∈ On ∧ 9o ≠ ∅) → ∃𝑡 ∈ On ∃𝑣 ∈ 9o ((9o ·o 𝑡) +o 𝑣) = 𝑤)
11	6, 9, 10	mp3an13 1481	. . . . . . 7 ⊢ (𝑤 ∈ On → ∃𝑡 ∈ On ∃𝑣 ∈ 9o ((9o ·o 𝑡) +o 𝑣) = 𝑤)
12		j0iso 35955	. . . . . . . . . . 11 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
13		isof1o 7335	. . . . . . . . . . 11 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)–1-1-onto→On)
14		f1ofo 6837	. . . . . . . . . . 11 ⊢ (𝐽0:(On × On)–1-1-onto→On → 𝐽0:(On × On)–onto→On)
15	12, 13, 14	mp2b 10	. . . . . . . . . 10 ⊢ 𝐽0:(On × On)–onto→On
16		foelcdmi 6951	. . . . . . . . . 10 ⊢ ((𝐽0:(On × On)–onto→On ∧ 𝑡 ∈ On) → ∃𝑢 ∈ (On × On)(𝐽0‘𝑢) = 𝑡)
17	15, 16	mpan 703	. . . . . . . . 9 ⊢ (𝑡 ∈ On → ∃𝑢 ∈ (On × On)(𝐽0‘𝑢) = 𝑡)
18		oveq2 7432	. . . . . . . . . . . . . . 15 ⊢ ((𝐽0‘𝑢) = 𝑡 → (9o ·o (𝐽0‘𝑢)) = (9o ·o 𝑡))
19	18	oveq1d 7439	. . . . . . . . . . . . . 14 ⊢ ((𝐽0‘𝑢) = 𝑡 → ((9o ·o (𝐽0‘𝑢)) +o 𝑣) = ((9o ·o 𝑡) +o 𝑣))
20	19	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ ((𝐽0‘𝑢) = 𝑡 → (𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣) ↔ 𝑤 = ((9o ·o 𝑡) +o 𝑣)))
21		eqcom 2767	. . . . . . . . . . . . 13 ⊢ (𝑤 = ((9o ·o 𝑡) +o 𝑣) ↔ ((9o ·o 𝑡) +o 𝑣) = 𝑤)
22	20, 21	bitrdi 290	. . . . . . . . . . . 12 ⊢ ((𝐽0‘𝑢) = 𝑡 → (𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣) ↔ ((9o ·o 𝑡) +o 𝑣) = 𝑤))
23	22	rexbidv 3186	. . . . . . . . . . 11 ⊢ ((𝐽0‘𝑢) = 𝑡 → (∃𝑣 ∈ 9o 𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣) ↔ ∃𝑣 ∈ 9o ((9o ·o 𝑡) +o 𝑣) = 𝑤))
24	23	biimprcd 253	. . . . . . . . . 10 ⊢ (∃𝑣 ∈ 9o ((9o ·o 𝑡) +o 𝑣) = 𝑤 → ((𝐽0‘𝑢) = 𝑡 → ∃𝑣 ∈ 9o 𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣)))
25	24	reximdv 3177	. . . . . . . . 9 ⊢ (∃𝑣 ∈ 9o ((9o ·o 𝑡) +o 𝑣) = 𝑤 → (∃𝑢 ∈ (On × On)(𝐽0‘𝑢) = 𝑡 → ∃𝑢 ∈ (On × On)∃𝑣 ∈ 9o 𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣)))
26	17, 25	mpan9 516	. . . . . . . 8 ⊢ ((𝑡 ∈ On ∧ ∃𝑣 ∈ 9o ((9o ·o 𝑡) +o 𝑣) = 𝑤) → ∃𝑢 ∈ (On × On)∃𝑣 ∈ 9o 𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣))
27	26	rexlimiva 3155	. . . . . . 7 ⊢ (∃𝑡 ∈ On ∃𝑣 ∈ 9o ((9o ·o 𝑡) +o 𝑣) = 𝑤 → ∃𝑢 ∈ (On × On)∃𝑣 ∈ 9o 𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣))
28	11, 27	syl 18	. . . . . 6 ⊢ (𝑤 ∈ On → ∃𝑢 ∈ (On × On)∃𝑣 ∈ 9o 𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣))
29		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
30		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
31	29, 30	op1std 8015	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑧) = 𝑢)
32	31	fveq2d 6894	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐽0‘(1st ‘𝑧)) = (𝐽0‘𝑢))
33	32	oveq2d 7440	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (9o ·o (𝐽0‘(1st ‘𝑧))) = (9o ·o (𝐽0‘𝑢)))
34	29, 30	op2ndd 8016	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑧) = 𝑣)
35	33, 34	oveq12d 7442	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧)) = ((9o ·o (𝐽0‘𝑢)) +o 𝑣))
36	35	eqeq2d 2771	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧)) ↔ 𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣)))
37	36	rexxp 5820	. . . . . 6 ⊢ (∃𝑧 ∈ ((On × On) × 9o)𝑤 = ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧)) ↔ ∃𝑢 ∈ (On × On)∃𝑣 ∈ 9o 𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣))
38	28, 37	sylibr 237	. . . . 5 ⊢ (𝑤 ∈ On → ∃𝑧 ∈ ((On × On) × 9o)𝑤 = ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧)))
39	5, 38	mpgbir 1831	. . . 4 ⊢ On ⊆ {𝑤 ∣ ∃𝑧 ∈ ((On × On) × 9o)𝑤 = ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧))}
40		dfj2 35960	. . . . 5 ⊢ 𝐽 = (𝑧 ∈ ((On × On) × 9o) ↦ ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧)))
41	40	rnmpt 5940	. . . 4 ⊢ ran 𝐽 = {𝑤 ∣ ∃𝑧 ∈ ((On × On) × 9o)𝑤 = ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧))}
42	39, 41	sseqtrri 3958	. . 3 ⊢ On ⊆ ran 𝐽
43	4, 42	eqssi 3925	. 2 ⊢ ran 𝐽 = On
44		dff1o5 6839	. 2 ⊢ (𝐽:((On × On) × 9o)–1-1-onto→On ↔ (𝐽:((On × On) × 9o)–1-1→On ∧ ran 𝐽 = On))
45	1, 43, 44	mpbir2an 724	1 ⊢ 𝐽:((On × On) × 9o)–1-1-onto→On

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∃wrex 3086   ⊆ wss 3896  ∅c0 4277  ⟨cop 4834   E cep 5550   × cxp 5649  ran crn 5652  Oncon0 6365  suc csuc 6367  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542  ‘cfv 6543   Isom wiso 6544  (class class class)co 7424  1^st c1st 8003  2^nd c2nd 8004   +_o coa 8481   ·_o comu 8482  8_oc8o 35832  9_oc9o 35833  𝑅₀cr0 35932  𝐽₀ccj0 35933  𝐽cj 35934

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 5327  ax-pr 5391  ax-un 7755

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-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-mpt3 7688  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-2o 8485  df-3o 8486  df-4o 8487  df-oadd 8488  df-omul 8489  df-oi 9511  df-5o 35834  df-6o 35835  df-7o 35836  df-8o 35837  df-9o 35838  df-lexo 35940  df-r0 35941  df-j0 35942  df-j 35943

This theorem is used by:  dfk12  35963  dfk22  35964  dfk32  35965  k1on  35966  k2on  35967  k39o  35968  jk1k2k3  35969