Theorem j00smo 36005

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 7446	. . . 4 ⊢ (∅𝐽0𝑥) = (𝐽0‘⟨∅, 𝑥⟩)
3		0elon 6436	. . . . 5 ⊢ ∅ ∈ On
4		j0iso 36004	. . . . . . 7 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
5		isof1o 7354	. . . . . . 7 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)⟶1-1ontoOn)
6		f1of 6846	. . . . . . 7 ⊢ (𝐽0:(On × On)⟶1-1ontoOn → 𝐽0:(On × On)⟶On)
7	4, 5, 6	mp2b 10	. . . . . 6 ⊢ 𝐽0:(On × On)⟶On
8	7	fovcl 7571	. . . . 5 ⊢ ((∅ ∈ On ∧ 𝑥 ∈ On) → (∅𝐽0𝑥) ∈ On)
9	3, 8	mpan 703	. . . 4 ⊢ (𝑥 ∈ On → (∅𝐽0𝑥) ∈ On)
10	2, 9	eqeltrrid 2869	. . 3 ⊢ (𝑥 ∈ On → (𝐽0‘⟨∅, 𝑥⟩) ∈ On)
11	1, 10	fmpti 7136	. 2 ⊢ 𝐹:On⟶On
12		ordon 7815	. 2 ⊢ Ord On
13		0un 4349	. . . . 5 ⊢ (∅ ∪ 𝑦) = 𝑦
14		0un 4349	. . . . 5 ⊢ (∅ ∪ 𝑧) = 𝑧
15	13, 14	eleq12i 2857	. . . 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 36001	. . . . 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 5692	. . . . . 6 ⊢ ((∅ ∈ On ∧ 𝑦 ∈ On) → ⟨∅, 𝑦⟩ ∈ (On × On))
24	3, 23	mpan 703	. . . . 5 ⊢ (𝑦 ∈ On → ⟨∅, 𝑦⟩ ∈ (On × On))
25		opelxpi 5692	. . . . . 6 ⊢ ((∅ ∈ On ∧ 𝑧 ∈ On) → ⟨∅, 𝑧⟩ ∈ (On × On))
26	3, 25	mpan 703	. . . . 5 ⊢ (𝑧 ∈ On → ⟨∅, 𝑧⟩ ∈ (On × On))
27		isorel 7357	. . . . . 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 6921	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
31	30	epeli 5557	. . . . 5 ⊢ ((𝐹‘𝑦) E (𝐹‘𝑧) ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑧))
32		opeq2 4844	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
33	32	fveq2d 6912	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐽0‘⟨∅, 𝑥⟩) = (𝐽0‘⟨∅, 𝑦⟩))
34		fvex 6921	. . . . . . 7 ⊢ (𝐽0‘⟨∅, 𝑦⟩) ∈ V
35	33, 1, 34	fvmpt 7016	. . . . . 6 ⊢ (𝑦 ∈ On → (𝐹‘𝑦) = (𝐽0‘⟨∅, 𝑦⟩))
36		opeq2 4844	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑧⟩)
37	36	fveq2d 6912	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝐽0‘⟨∅, 𝑥⟩) = (𝐽0‘⟨∅, 𝑧⟩))
38		fvex 6921	. . . . . . 7 ⊢ (𝐽0‘⟨∅, 𝑧⟩) ∈ V
39	37, 1, 38	fvmpt 7016	. . . . . 6 ⊢ (𝑧 ∈ On → (𝐹‘𝑧) = (𝐽0‘⟨∅, 𝑧⟩))
40	35, 39	breqan12d 5122	. . . . 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 6743	. 2 ⊢ dom 𝐹 = On
45	11, 12, 43, 44	issmo 8384	1 ⊢ Smo 𝐹

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ∪ cun 4152  ∅c0 4281  ⟨cop 4838   class class class wbr 5106   ↦ cmpt 5189   E cep 5554   × cxp 5653  Oncon0 6380  ⟶wf 6554  ⟶1-1ontowf1o 6557  ‘cfv 6558   Isom wiso 6559  (class class class)co 7443  Smo wsmo 8381  LexOrdclexo 35980  𝑅₀cr0 35981  𝐽₀ccj0 35982

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 7775

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 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-se 5609  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 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-smo 8382  df-recs 8407  df-oi 9531  df-lexo 35989  df-r0 35990  df-j0 35991

This theorem is used by:  maxssj0  36006