Theorem jk1k2k3 35969

Description: Identity law for the 𝐽 function. Proposition 15.8 of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 11-Oct-2026.)

Assertion

Ref	Expression
jk1k2k3	⊢ (𝐴 ∈ On → 𝐴 = (𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩))

Proof of Theorem jk1k2k3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		jf1o 35962	. . . . . 6 ⊢ 𝐽:((On × On) × 9o)–1-1-onto→On
2		f1ocnvdm 7297	. . . . . 6 ⊢ ((𝐽:((On × On) × 9o)–1-1-onto→On ∧ 𝐴 ∈ On) → (◡𝐽‘𝐴) ∈ ((On × On) × 9o))
3	1, 2	mpan 703	. . . . 5 ⊢ (𝐴 ∈ On → (◡𝐽‘𝐴) ∈ ((On × On) × 9o))
4		2fveq3 6895	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (1st ‘(◡𝐽‘𝑥)) = (1st ‘(◡𝐽‘𝐴)))
5	4	fveq2d 6894	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (1st ‘(1st ‘(◡𝐽‘𝑥))) = (1st ‘(1st ‘(◡𝐽‘𝐴))))
6		dfk12 35963	. . . . . . . 8 ⊢ 𝐾1 = (𝑥 ∈ On ↦ (1st ‘(1st ‘(◡𝐽‘𝑥))))
7		fvex 6903	. . . . . . . 8 ⊢ (1st ‘(1st ‘(◡𝐽‘𝐴))) ∈ V
8	5, 6, 7	fvmpt 6998	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐾1‘𝐴) = (1st ‘(1st ‘(◡𝐽‘𝐴))))
9	8	eqcomd 2766	. . . . . 6 ⊢ (𝐴 ∈ On → (1st ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾1‘𝐴))
10	4	fveq2d 6894	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (2nd ‘(1st ‘(◡𝐽‘𝑥))) = (2nd ‘(1st ‘(◡𝐽‘𝐴))))
11		dfk22 35964	. . . . . . . 8 ⊢ 𝐾2 = (𝑥 ∈ On ↦ (2nd ‘(1st ‘(◡𝐽‘𝑥))))
12		fvex 6903	. . . . . . . 8 ⊢ (2nd ‘(1st ‘(◡𝐽‘𝐴))) ∈ V
13	10, 11, 12	fvmpt 6998	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐾2‘𝐴) = (2nd ‘(1st ‘(◡𝐽‘𝐴))))
14	13	eqcomd 2766	. . . . . 6 ⊢ (𝐴 ∈ On → (2nd ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾2‘𝐴))
15		2fveq3 6895	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (2nd ‘(◡𝐽‘𝑥)) = (2nd ‘(◡𝐽‘𝐴)))
16		dfk32 35965	. . . . . . . 8 ⊢ 𝐾3 = (𝑥 ∈ On ↦ (2nd ‘(◡𝐽‘𝑥)))
17		fvex 6903	. . . . . . . 8 ⊢ (2nd ‘(◡𝐽‘𝐴)) ∈ V
18	15, 16, 17	fvmpt 6998	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐾3‘𝐴) = (2nd ‘(◡𝐽‘𝐴)))
19	18	eqcomd 2766	. . . . . 6 ⊢ (𝐴 ∈ On → (2nd ‘(◡𝐽‘𝐴)) = (𝐾3‘𝐴))
20	9, 14, 19	3jca 1146	. . . . 5 ⊢ (𝐴 ∈ On → ((1st ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾1‘𝐴) ∧ (2nd ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾2‘𝐴) ∧ (2nd ‘(◡𝐽‘𝐴)) = (𝐾3‘𝐴)))
21	3, 20	jca 521	. . . 4 ⊢ (𝐴 ∈ On → ((◡𝐽‘𝐴) ∈ ((On × On) × 9o) ∧ ((1st ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾1‘𝐴) ∧ (2nd ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾2‘𝐴) ∧ (2nd ‘(◡𝐽‘𝐴)) = (𝐾3‘𝐴))))
22		k1on 35966	. . . . 5 ⊢ (𝐾1‘𝐴) ∈ On
23		k2on 35967	. . . . 5 ⊢ (𝐾2‘𝐴) ∈ On
24		k39o 35968	. . . . 5 ⊢ (𝐾3‘𝐴) ∈ 9o
25		el2xptp0 8051	. . . . 5 ⊢ (((𝐾1‘𝐴) ∈ On ∧ (𝐾2‘𝐴) ∈ On ∧ (𝐾3‘𝐴) ∈ 9o) → (((◡𝐽‘𝐴) ∈ ((On × On) × 9o) ∧ ((1st ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾1‘𝐴) ∧ (2nd ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾2‘𝐴) ∧ (2nd ‘(◡𝐽‘𝐴)) = (𝐾3‘𝐴))) ↔ (◡𝐽‘𝐴) = ⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩))
26	22, 23, 24, 25	mp3an 1490	. . . 4 ⊢ (((◡𝐽‘𝐴) ∈ ((On × On) × 9o) ∧ ((1st ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾1‘𝐴) ∧ (2nd ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾2‘𝐴) ∧ (2nd ‘(◡𝐽‘𝐴)) = (𝐾3‘𝐴))) ↔ (◡𝐽‘𝐴) = ⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩)
27	21, 26	sylib 221	. . 3 ⊢ (𝐴 ∈ On → (◡𝐽‘𝐴) = ⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩)
28	27, 3	eqeltrrd 2861	. . . 4 ⊢ (𝐴 ∈ On → ⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩ ∈ ((On × On) × 9o))
29		f1ocnvfvb 7291	. . . . 5 ⊢ ((𝐽:((On × On) × 9o)–1-1-onto→On ∧ ⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩ ∈ ((On × On) × 9o) ∧ 𝐴 ∈ On) → ((𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩) = 𝐴 ↔ (◡𝐽‘𝐴) = ⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩))
30	1, 29	mp3an1 1477	. . . 4 ⊢ ((⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩ ∈ ((On × On) × 9o) ∧ 𝐴 ∈ On) → ((𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩) = 𝐴 ↔ (◡𝐽‘𝐴) = ⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩))
31	28, 30	mpancom 701	. . 3 ⊢ (𝐴 ∈ On → ((𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩) = 𝐴 ↔ (◡𝐽‘𝐴) = ⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩))
32	27, 31	mpbird 260	. 2 ⊢ (𝐴 ∈ On → (𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩) = 𝐴)
33	32	eqcomd 2766	1 ⊢ (𝐴 ∈ On → 𝐴 = (𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ⟨cotp 4857   × cxp 5649  ^◡ccnv 5650  Oncon0 6365  –1-1-onto→wf1o 6542  ‘cfv 6543  1^st c1st 8003  2^nd c2nd 8004  9_oc9o 35833  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:  j0le  35970  k3n0j0lt  35971