Theorem dfj2 36050

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 36033	. 2 ⊢ 𝐽 = (𝑦 ∈ On, 𝑧 ∈ On, 𝑛 ∈ 9o ↦ ((9o ·o (𝐽0‘⟨𝑦, 𝑧⟩)) +o 𝑛))
2		df-ot 4864	. . . . . . . 8 ⊢ ⟨𝑦, 𝑧, 𝑛⟩ = ⟨⟨𝑦, 𝑧⟩, 𝑛⟩
3	2	eqeq2i 2776	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑛⟩ ↔ 𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑛⟩)
4		opex 5439	. . . . . . . 8 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
5		vex 3458	. . . . . . . 8 ⊢ 𝑛 ∈ V
6	4, 5	op1std 8045	. . . . . . 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 6920	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑛⟩ → (𝐽0‘(1st ‘𝑥)) = (𝐽0‘⟨𝑦, 𝑧⟩))
9	8	oveq2d 7467	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑛⟩ → (9o ·o (𝐽0‘(1st ‘𝑥))) = (9o ·o (𝐽0‘⟨𝑦, 𝑧⟩)))
10	4, 5	op2ndd 8046	. . . . 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 7469	. . 3 ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑛⟩ → ((9o ·o (𝐽0‘(1st ‘𝑥))) +o (2nd ‘𝑥)) = ((9o ·o (𝐽0‘⟨𝑦, 𝑧⟩)) +o 𝑛))
13	12	mpt3mpt 7717	. 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 2789	1 ⊢ 𝐽 = (𝑥 ∈ ((On × On) × 9o) ↦ ((9o ·o (𝐽0‘(1st ‘𝑥))) +o (2nd ‘𝑥)))

This proof depends on syntax axioms:   = wceq 1570  ⟨cop 4840  ⟨cotp 4863   ↦ cmpt 5192   × cxp 5656  Oncon0 6388  ‘cfv 6566  (class class class)co 7451   ∈ cmpt3 7715  1^st c1st 8033  2^nd c2nd 8034   +_o coa 8511   ·_o comu 8512  9_oc9o 35898  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-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-ot 4864  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-mpt3 7716  df-1st 8035  df-2nd 8036  df-j 36033

This theorem is used by:  jf1  36051  jf1o  36052