Theorem jf1o 36052

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)⟶1-1ontoOn

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 36051	. 2 ⊢ 𝐽:((On × On) × 9o)⟶1-1On
2		f1f 6808	. . . 4 ⊢ (𝐽:((On × On) × 9o)⟶1-1On → 𝐽:((On × On) × 9o)⟶On)
3		frn 6747	. . . 4 ⊢ (𝐽:((On × On) × 9o)⟶On → ran 𝐽 ⊆ On)
4	1, 2, 3	mp2b 10	. . 3 ⊢ ran 𝐽 ⊆ On
5		ssab 3991	. . . . 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 35908	. . . . . . . 8 ⊢ 9o ∈ On
7		df-9o 35903	. . . . . . . . 9 ⊢ 9o = suc 8o
8		nsuceq0 6474	. . . . . . . . 9 ⊢ suc 8o ≠ ∅
9	7, 8	eqnetri 3029	. . . . . . . 8 ⊢ 9o ≠ ∅
10		omeulem1 8628	. . . . . . . 8 ⊢ ((9o ∈ On ∧ 𝑤 ∈ On ∧ 9o ≠ ∅) → ∃𝑡 ∈ On ∃𝑣 ∈ 9o ((9o ·o 𝑡) +o 𝑣) = 𝑤)
11	6, 9, 10	mp3an13 1482	. . . . . . 7 ⊢ (𝑤 ∈ On → ∃𝑡 ∈ On ∃𝑣 ∈ 9o ((9o ·o 𝑡) +o 𝑣) = 𝑤)
12		j0iso 36045	. . . . . . . . . . 11 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
13		isof1o 7362	. . . . . . . . . . 11 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)⟶1-1ontoOn)
14		f1ofo 6862	. . . . . . . . . . 11 ⊢ (𝐽0:(On × On)⟶1-1ontoOn → 𝐽0:(On × On)⟶ontoOn)
15	12, 13, 14	mp2b 10	. . . . . . . . . 10 ⊢ 𝐽0:(On × On)⟶ontoOn
16		foelcdmi 6977	. . . . . . . . . 10 ⊢ ((𝐽0:(On × On)⟶ontoOn ∧ 𝑡 ∈ On) → ∃𝑢 ∈ (On × On)(𝐽0‘𝑢) = 𝑡)
17	15, 16	mpan 703	. . . . . . . . 9 ⊢ (𝑡 ∈ On → ∃𝑢 ∈ (On × On)(𝐽0‘𝑢) = 𝑡)
18		oveq2 7459	. . . . . . . . . . . . . . 15 ⊢ ((𝐽0‘𝑢) = 𝑡 → (9o ·o (𝐽0‘𝑢)) = (9o ·o 𝑡))
19	18	oveq1d 7466	. . . . . . . . . . . . . 14 ⊢ ((𝐽0‘𝑢) = 𝑡 → ((9o ·o (𝐽0‘𝑢)) +o 𝑣) = ((9o ·o 𝑡) +o 𝑣))
20	19	eqeq2d 2774	. . . . . . . . . . . . 13 ⊢ ((𝐽0‘𝑢) = 𝑡 → (𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣) ↔ 𝑤 = ((9o ·o 𝑡) +o 𝑣)))
21		eqcom 2770	. . . . . . . . . . . . 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 3190	. . . . . . . . . . 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 3181	. . . . . . . . 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 3159	. . . . . . 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 3458	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
30		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
31	29, 30	op1std 8045	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑧) = 𝑢)
32	31	fveq2d 6920	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐽0‘(1st ‘𝑧)) = (𝐽0‘𝑢))
33	32	oveq2d 7467	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (9o ·o (𝐽0‘(1st ‘𝑧))) = (9o ·o (𝐽0‘𝑢)))
34	29, 30	op2ndd 8046	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑧) = 𝑣)
35	33, 34	oveq12d 7469	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧)) = ((9o ·o (𝐽0‘𝑢)) +o 𝑣))
36	35	eqeq2d 2774	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧)) ↔ 𝑤 = ((9o ·o (𝐽0‘𝑢)) +o 𝑣)))
37	36	rexxp 5829	. . . . . 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 1832	. . . 4 ⊢ On ⊆ {𝑤 ∣ ∃𝑧 ∈ ((On × On) × 9o)𝑤 = ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧))}
40		dfj2 36050	. . . . 5 ⊢ 𝐽 = (𝑧 ∈ ((On × On) × 9o) ↦ ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧)))
41	40	rnmpt 5951	. . . 4 ⊢ ran 𝐽 = {𝑤 ∣ ∃𝑧 ∈ ((On × On) × 9o)𝑤 = ((9o ·o (𝐽0‘(1st ‘𝑧))) +o (2nd ‘𝑧))}
42	39, 41	sseqtrri 3962	. . 3 ⊢ On ⊆ ran 𝐽
43	4, 42	eqssi 3929	. 2 ⊢ ran 𝐽 = On
44		dff1o5 6864	. 2 ⊢ (𝐽:((On × On) × 9o)⟶1-1ontoOn ↔ (𝐽:((On × On) × 9o)⟶1-1On ∧ ran 𝐽 = On))
45	1, 43, 44	mpbir2an 724	1 ⊢ 𝐽:((On × On) × 9o)⟶1-1ontoOn

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281  ⟨cop 4840   E cep 5557   × cxp 5656  ran crn 5659  Oncon0 6388  suc csuc 6390  ⟶wf 6562  ⟶1-1wf1 6563  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565  ‘cfv 6566   Isom wiso 6567  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034   +_o coa 8511   ·_o comu 8512  8_oc8o 35897  9_oc9o 35898  𝑅₀cr0 36022  𝐽₀ccj0 36023  𝐽cj 36024

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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-rmo 3369  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 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-mpt3 7716  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-3o 8516  df-4o 8517  df-oadd 8518  df-omul 8519  df-oi 9541  df-5o 35899  df-6o 35900  df-7o 35901  df-8o 35902  df-9o 35903  df-lexo 36030  df-r0 36031  df-j0 36032  df-j 36033

This theorem is used by:  dfk12  36053  dfk22  36054  dfk32  36055  k1on  36056  k2on  36057  k39o  36058  jk1k2k3  36059