Theorem j00smo 35937

Description: 𝐽₀ is strictly monotonic over its second argument when its first argument is zero. (Contributed by BTernaryTau, 19-Sep-2026.)

Hypothesis

Ref	Expression
j00smo.1	⊢ 𝐹 = (𝑥 ∈ On ↦ (𝐽0‘⟨∅, 𝑥⟩))

Assertion

Ref	Expression
j00smo	⊢ Smo 𝐹

Proof of Theorem j00smo

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		j00smo.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ On ↦ (𝐽0‘⟨∅, 𝑥⟩))
2		df-ov 7423	. . . 4 ⊢ (∅𝐽0𝑥) = (𝐽0‘⟨∅, 𝑥⟩)
3		0elon 6418	. . . . 5 ⊢ ∅ ∈ On
4		j0iso 35936	. . . . . . 7 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
5		isof1o 7331	. . . . . . 7 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)–1-1-onto→On)
6		f1of 6824	. . . . . . 7 ⊢ (𝐽0:(On × On)–1-1-onto→On → 𝐽0:(On × On)⟶On)
7	4, 5, 6	mp2b 10	. . . . . 6 ⊢ 𝐽0:(On × On)⟶On
8	7	fovcl 7548	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝑥 ∈ On) → (∅𝐽0𝑥) ∈ On)
9	3, 8	mpan 703	. . . 4 ⊢ (𝑥 ∈ On → (∅𝐽0𝑥) ∈ On)
10	2, 9	eqeltrrid 2865	. . 3 ⊢ (𝑥 ∈ On → (𝐽0‘⟨∅, 𝑥⟩) ∈ On)
11	1, 10	fmpti 7112	. 2 ⊢ 𝐹:On⟶On
12		ordon 7791	. 2 ⊢ Ord On
13		0un 4345	. . . . 5 ⊢ (∅ ∪ 𝑦) = 𝑦
14		0un 4345	. . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧
15	13, 14	eleq12i 2853	. . . 4 ⊢ ((∅ ∪ 𝑦) ∈ (∅ ∪ 𝑧) ↔ 𝑦 ∈ 𝑧)
16		orc 881	. . . . 5 ⊢ ((∅ ∪ 𝑦) ∈ (∅ ∪ 𝑧) → ((∅ ∪ 𝑦) ∈ (∅ ∪ 𝑧) ∨ ((∅ ∪ 𝑦) = (∅ ∪ 𝑧) ∧ ⟨∅, 𝑦⟩LexOrd⟨∅, 𝑧⟩)))
17	3	a1i 11	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ∅ ∈ On)
18		simpl 488	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → 𝑦 ∈ On)
19		simpr 490	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → 𝑧 ∈ On)
20	17, 18, 17, 19	brr0d 35933	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (⟨∅, 𝑦⟩𝑅0⟨∅, 𝑧⟩ ↔ ((∅ ∪ 𝑦) ∈ (∅ ∪ 𝑧) ∨ ((∅ ∪ 𝑦) = (∅ ∪ 𝑧) ∧ ⟨∅, 𝑦⟩LexOrd⟨∅, 𝑧⟩))))
21	16, 20	imbitrrid 249	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((∅ ∪ 𝑦) ∈ (∅ ∪ 𝑧) → ⟨∅, 𝑦⟩𝑅0⟨∅, 𝑧⟩))
22	15, 21	biimtrrid 246	. . 3 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 → ⟨∅, 𝑦⟩𝑅0⟨∅, 𝑧⟩))
23		opelxpi 5687	. . . . . 6 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → ⟨∅, 𝑦⟩ ∈ (On × On))
24	3, 23	mpan 703	. . . . 5 ⊢ (𝑦 ∈ On → ⟨∅, 𝑦⟩ ∈ (On × On))
25		opelxpi 5687	. . . . . 6 ⊢ ((∅ ∈ On ∧ 𝑧 ∈ On) → ⟨∅, 𝑧⟩ ∈ (On × On))
26	3, 25	mpan 703	. . . . 5 ⊢ (𝑧 ∈ On → ⟨∅, 𝑧⟩ ∈ (On × On))
27		isorel 7334	. . . . . 6 ⊢ ((𝐽0 Isom 𝑅0, E ((On × On), On) ∧ (⟨∅, 𝑦⟩ ∈ (On × On) ∧ ⟨∅, 𝑧⟩ ∈ (On × On))) → (⟨∅, 𝑦⟩𝑅0⟨∅, 𝑧⟩ ↔ (𝐽0‘⟨∅, 𝑦⟩) E (𝐽0‘⟨∅, 𝑧⟩)))
28	4, 27	mpan 703	. . . . 5 ⊢ ((⟨∅, 𝑦⟩ ∈ (On × On) ∧ ⟨∅, 𝑧⟩ ∈ (On × On)) → (⟨∅, 𝑦⟩𝑅0⟨∅, 𝑧⟩ ↔ (𝐽0‘⟨∅, 𝑦⟩) E (𝐽0‘⟨∅, 𝑧⟩)))
29	24, 26, 28	syl2an 608	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (⟨∅, 𝑦⟩𝑅0⟨∅, 𝑧⟩ ↔ (𝐽0‘⟨∅, 𝑦⟩) E (𝐽0‘⟨∅, 𝑧⟩)))
30		fvex 6898	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
31	30	epeli 5552	. . . . 5 ⊢ ((𝐹‘𝑦) E (𝐹‘𝑧) ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑧))
32		opeq2 4833	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
33	32	fveq2d 6889	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐽0‘⟨∅, 𝑥⟩) = (𝐽0‘⟨∅, 𝑦⟩))
34		fvex 6898	. . . . . . 7 ⊢ (𝐽0‘⟨∅, 𝑦⟩) ∈ V
35	33, 1, 34	fvmpt 6993	. . . . . 6 ⊢ (𝑦 ∈ On → (𝐹‘𝑦) = (𝐽0‘⟨∅, 𝑦⟩))
36		opeq2 4833	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑧⟩)
37	36	fveq2d 6889	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝐽0‘⟨∅, 𝑥⟩) = (𝐽0‘⟨∅, 𝑧⟩))
38		fvex 6898	. . . . . . 7 ⊢ (𝐽0‘⟨∅, 𝑧⟩) ∈ V
39	37, 1, 38	fvmpt 6993	. . . . . 6 ⊢ (𝑧 ∈ On → (𝐹‘𝑧) = (𝐽0‘⟨∅, 𝑧⟩))
40	35, 39	breqan12d 5118	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((𝐹‘𝑦) E (𝐹‘𝑧) ↔ (𝐽0‘⟨∅, 𝑦⟩) E (𝐽0‘⟨∅, 𝑧⟩)))
41	31, 40	bitr3id 288	. . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ↔ (𝐽0‘⟨∅, 𝑦⟩) E (𝐽0‘⟨∅, 𝑧⟩)))
42	29, 41	bitr4d 285	. . 3 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (⟨∅, 𝑦⟩𝑅0⟨∅, 𝑧⟩ ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑧)))
43	22, 42	sylibd 242	. 2 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ∈ 𝑧 → (𝐹‘𝑦) ∈ (𝐹‘𝑧)))
44	11	fdmi 6721	. 2 ⊢ dom 𝐹 = On
45	11, 12, 43, 44	issmo 8357	1 ⊢ Smo 𝐹

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ∪ cun 4149  ∅c0 4278  ⟨cop 4589   class class class wbr 5102   ↦ cmpt 5185   E cep 5549   × cxp 5648  Oncon0 6362  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539  (class class class)co 7420  Smo wsmo 8354  LexOrdclexo 35912  𝑅₀cr0 35913  𝐽₀ccj0 35914

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

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 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-se 5604  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-isom 6547  df-riota 7377  df-ov 7423  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-smo 8355  df-recs 8380  df-oi 9505  df-lexo 35921  df-r0 35922  df-j0 35923

This theorem is used by:  maxssj0  35938