Theorem dfj2 35960

Description: Alternate definition of 𝐽 as a one-argument function. (Contributed by BTernaryTau, 4-Oct-2026.)

Assertion

Ref	Expression
dfj2	⊢ 𝐽 = (𝑥 ∈ ((On × On) × 9o) ↦ ((9o ·o (𝐽0‘(1st ‘𝑥))) +o (2nd ‘𝑥)))

Proof of Theorem dfj2

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

Step	Hyp	Ref	Expression
1		df-j 35943	. 2 ⊢ 𝐽 = (𝑦 ∈ On, 𝑧 ∈ On, 𝑛 ∈ 9o ↦ ((9o ·o (𝐽0‘⟨𝑦, 𝑧⟩)) +o 𝑛))
2		df-ot 4858	. . . . . . . 8 ⊢ ⟨𝑦, 𝑧, 𝑛⟩ = ⟨⟨𝑦, 𝑧⟩, 𝑛⟩
3	2	eqeq2i 2773	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑛⟩ ↔ 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑛⟩)
4		opex 5432	. . . . . . . 8 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
5		vex 3454	. . . . . . . 8 ⊢ 𝑛 ∈ V
6	4, 5	op1std 8015	. . . . . . 7 ⊢ (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑛⟩ → (1st ‘𝑥) = ⟨𝑦, 𝑧⟩)
7	3, 6	sylbi 220	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑛⟩ → (1st ‘𝑥) = ⟨𝑦, 𝑧⟩)
8	7	fveq2d 6894	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑛⟩ → (𝐽0‘(1st ‘𝑥)) = (𝐽0‘⟨𝑦, 𝑧⟩))
9	8	oveq2d 7440	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑛⟩ → (9o ·o (𝐽0‘(1st ‘𝑥))) = (9o ·o (𝐽0‘⟨𝑦, 𝑧⟩)))
10	4, 5	op2ndd 8016	. . . . 5 ⊢ (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑛⟩ → (2nd ‘𝑥) = 𝑛)
11	3, 10	sylbi 220	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑛⟩ → (2nd ‘𝑥) = 𝑛)
12	9, 11	oveq12d 7442	. . 3 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑛⟩ → ((9o ·o (𝐽0‘(1st ‘𝑥))) +o (2nd ‘𝑥)) = ((9o ·o (𝐽0‘⟨𝑦, 𝑧⟩)) +o 𝑛))
13	12	mpt3mpt 7689	. 2 ⊢ (𝑥 ∈ ((On × On) × 9o) ↦ ((9o ·o (𝐽0‘(1st ‘𝑥))) +o (2nd ‘𝑥))) = (𝑦 ∈ On, 𝑧 ∈ On, 𝑛 ∈ 9o ↦ ((9o ·o (𝐽0‘⟨𝑦, 𝑧⟩)) +o 𝑛))
14	1, 13	eqtr4i 2786	1 ⊢ 𝐽 = (𝑥 ∈ ((On × On) × 9o) ↦ ((9o ·o (𝐽0‘(1st ‘𝑥))) +o (2nd ‘𝑥)))

This proof depends on syntax axioms:   = wceq 1569  ⟨cop 4834  ⟨cotp 4857   ↦ cmpt 5185   × cxp 5649  Oncon0 6365  ‘cfv 6543  (class class class)co 7424   ∈ cmpt3 7687  1^st c1st 8003  2^nd c2nd 8004   +_o coa 8481   ·_o comu 8482  9_oc9o 35833  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-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-mpt3 7688  df-1st 8005  df-2nd 8006  df-j 35943

This theorem is used by:  jf1  35961  jf1o  35962