Theorem j0le 35970

Description: An upper bound on the value of 𝐽₀. (Contributed by BTernaryTau, 12-Oct-2026.)

Assertion

Ref	Expression
j0le	⊢ (𝐴 ∈ On → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ 𝐴)

Proof of Theorem j0le

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		j0iso 35955	. . . . 5 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
2		isof1o 7335	. . . . 5 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)–1-1-onto→On)
3		f1of 6829	. . . . 5 ⊢ (𝐽0:(On × On)–1-1-onto→On → 𝐽0:(On × On)⟶On)
4	1, 2, 3	mp2b 10	. . . 4 ⊢ 𝐽0:(On × On)⟶On
5		0elon 6421	. . . 4 ⊢ ∅ ∈ On
6	4, 5	f0cli 7103	. . 3 ⊢ (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ On
7		9on 35843	. . 3 ⊢ 9o ∈ On
8		k39o 35968	. . . . . 6 ⊢ (𝐾3‘𝐴) ∈ 9o
9	8	ne0ii 4289	. . . . 5 ⊢ 9o ≠ ∅
10		on0eln0 6423	. . . . . 6 ⊢ (9o ∈ On → (∅ ∈ 9o ↔ 9o ≠ ∅))
11	7, 10	ax-mp 5	. . . . 5 ⊢ (∅ ∈ 9o ↔ 9o ≠ ∅)
12	9, 11	mpbir 234	. . . 4 ⊢ ∅ ∈ 9o
13		omword2 8590	. . . 4 ⊢ ((((𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ On ∧ 9o ∈ On) ∧ ∅ ∈ 9o) → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)))
14	12, 13	mpan2 704	. . 3 ⊢ (((𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ On ∧ 9o ∈ On) → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)))
15	6, 7, 14	mp2an 705	. 2 ⊢ (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩))
16		omcl 8552	. . . . 5 ⊢ ((9o ∈ On ∧ (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ On) → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ On)
17	7, 6, 16	mp2an 705	. . . 4 ⊢ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ On
18		onelon 6390	. . . . 5 ⊢ ((9o ∈ On ∧ (𝐾3‘𝐴) ∈ 9o) → (𝐾3‘𝐴) ∈ On)
19	7, 8, 18	mp2an 705	. . . 4 ⊢ (𝐾3‘𝐴) ∈ On
20		oaword1 8568	. . . 4 ⊢ (((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ∈ On ∧ (𝐾3‘𝐴) ∈ On) → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ⊆ ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)))
21	17, 19, 20	mp2an 705	. . 3 ⊢ (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ⊆ ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴))
22		jk1k2k3 35969	. . . 4 ⊢ (𝐴 ∈ On → 𝐴 = (𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩))
23		k1on 35966	. . . . . 6 ⊢ (𝐾1‘𝐴) ∈ On
24	23	a1i 11	. . . . 5 ⊢ (𝐴 ∈ On → (𝐾1‘𝐴) ∈ On)
25		k2on 35967	. . . . . 6 ⊢ (𝐾2‘𝐴) ∈ On
26	25	a1i 11	. . . . 5 ⊢ (𝐴 ∈ On → (𝐾2‘𝐴) ∈ On)
27	8	a1i 11	. . . . 5 ⊢ (𝐴 ∈ On → (𝐾3‘𝐴) ∈ 9o)
28		ovexd 7459	. . . . 5 ⊢ (𝐴 ∈ On → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)) ∈ V)
29		opeq1 4839	. . . . . . . . 9 ⊢ ((𝐾1‘𝐴) = 𝑥 → ⟨(𝐾1‘𝐴), 𝑦⟩ = ⟨𝑥, 𝑦⟩)
30	29	fveq2d 6894	. . . . . . . 8 ⊢ ((𝐾1‘𝐴) = 𝑥 → (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩) = (𝐽0‘⟨𝑥, 𝑦⟩))
31	30	oveq2d 7440	. . . . . . 7 ⊢ ((𝐾1‘𝐴) = 𝑥 → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)) = (9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)))
32	31	oveq1d 7439	. . . . . 6 ⊢ ((𝐾1‘𝐴) = 𝑥 → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)) +o 𝑛))
33	32	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐾1‘𝐴) = 𝑥) → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)) +o 𝑛))
34		opeq2 4840	. . . . . . . . 9 ⊢ ((𝐾2‘𝐴) = 𝑦 → ⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩ = ⟨(𝐾1‘𝐴), 𝑦⟩)
35	34	fveq2d 6894	. . . . . . . 8 ⊢ ((𝐾2‘𝐴) = 𝑦 → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) = (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩))
36	35	oveq2d 7440	. . . . . . 7 ⊢ ((𝐾2‘𝐴) = 𝑦 → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) = (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)))
37	36	oveq1d 7439	. . . . . 6 ⊢ ((𝐾2‘𝐴) = 𝑦 → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)) +o 𝑛))
38	37	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐾2‘𝐴) = 𝑦) → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), 𝑦⟩)) +o 𝑛))
39		oveq2 7432	. . . . . 6 ⊢ ((𝐾3‘𝐴) = 𝑛 → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)) = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o 𝑛))
40	39	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) = 𝑛) → ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)) = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o 𝑛))
41		df-j 35943	. . . . 5 ⊢ 𝐽 = (𝑥 ∈ On, 𝑦 ∈ On, 𝑛 ∈ 9o ↦ ((9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)) +o 𝑛))
42	24, 26, 27, 28, 33, 38, 40, 41	mpt3fvot2d 7694	. . . 4 ⊢ (𝐴 ∈ On → (𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩) = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)))
43	22, 42	eqtrd 2795	. . 3 ⊢ (𝐴 ∈ On → 𝐴 = ((9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) +o (𝐾3‘𝐴)))
44	21, 43	sseqtrrid 3952	. 2 ⊢ (𝐴 ∈ On → (9o ·o (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩)) ⊆ 𝐴)
45	15, 44	sstrid 3920	1 ⊢ (𝐴 ∈ On → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450   ⊆ wss 3896  ∅c0 4277  ⟨cop 4834  ⟨cotp 4857   E cep 5550   × cxp 5649  Oncon0 6365  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543   Isom wiso 6544  (class class class)co 7424   +_o coa 8481   ·_o comu 8482  9_oc9o 35833  𝑅₀cr0 35932  𝐽₀ccj0 35933  𝐽cj 35934  𝐾₁ck1 35935  𝐾₂ck2 35936  𝐾₃ck3 35937

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  df-k1 35944  df-k2 35945  df-k3 35946

This theorem is used by:  k1le  35972  k2le  35973