Theorem jk1k2k3 36059

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 36052	. . . . . 6 ⊢ 𝐽:((On × On) × 9o)⟶1-1ontoOn
2		f1ocnvdm 7324	. . . . . 6 ⊢ ((𝐽:((On × On) × 9o)⟶1-1ontoOn ∧ 𝐴 ∈ On) → (◡𝐽‘𝐴) ∈ ((On × On) × 9o))
3	1, 2	mpan 703	. . . . 5 ⊢ (𝐴 ∈ On → (◡𝐽‘𝐴) ∈ ((On × On) × 9o))
4		2fveq3 6921	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (1st ‘(◡𝐽‘𝑥)) = (1st ‘(◡𝐽‘𝐴)))
5	4	fveq2d 6920	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (1st ‘(1st ‘(◡𝐽‘𝑥))) = (1st ‘(1st ‘(◡𝐽‘𝐴))))
6		dfk12 36053	. . . . . . . 8 ⊢ 𝐾1 = (𝑥 ∈ On ↦ (1st ‘(1st ‘(◡𝐽‘𝑥))))
7		fvex 6929	. . . . . . . 8 ⊢ (1st ‘(1st ‘(◡𝐽‘𝐴))) ∈ V
8	5, 6, 7	fvmpt 7024	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐾1‘𝐴) = (1st ‘(1st ‘(◡𝐽‘𝐴))))
9	8	eqcomd 2769	. . . . . 6 ⊢ (𝐴 ∈ On → (1st ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾1‘𝐴))
10	4	fveq2d 6920	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (2nd ‘(1st ‘(◡𝐽‘𝑥))) = (2nd ‘(1st ‘(◡𝐽‘𝐴))))
11		dfk22 36054	. . . . . . . 8 ⊢ 𝐾2 = (𝑥 ∈ On ↦ (2nd ‘(1st ‘(◡𝐽‘𝑥))))
12		fvex 6929	. . . . . . . 8 ⊢ (2nd ‘(1st ‘(◡𝐽‘𝐴))) ∈ V
13	10, 11, 12	fvmpt 7024	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐾2‘𝐴) = (2nd ‘(1st ‘(◡𝐽‘𝐴))))
14	13	eqcomd 2769	. . . . . 6 ⊢ (𝐴 ∈ On → (2nd ‘(1st ‘(◡𝐽‘𝐴))) = (𝐾2‘𝐴))
15		2fveq3 6921	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (2nd ‘(◡𝐽‘𝑥)) = (2nd ‘(◡𝐽‘𝐴)))
16		dfk32 36055	. . . . . . . 8 ⊢ 𝐾3 = (𝑥 ∈ On ↦ (2nd ‘(◡𝐽‘𝑥)))
17		fvex 6929	. . . . . . . 8 ⊢ (2nd ‘(◡𝐽‘𝐴)) ∈ V
18	15, 16, 17	fvmpt 7024	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐾3‘𝐴) = (2nd ‘(◡𝐽‘𝐴)))
19	18	eqcomd 2769	. . . . . 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 36056	. . . . 5 ⊢ (𝐾1‘𝐴) ∈ On
23		k2on 36057	. . . . 5 ⊢ (𝐾2‘𝐴) ∈ On
24		k39o 36058	. . . . 5 ⊢ (𝐾3‘𝐴) ∈ 9o
25		el2xptp0 8081	. . . . 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 1491	. . . 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 2865	. . . 4 ⊢ (𝐴 ∈ On → ⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩ ∈ ((On × On) × 9o))
29		f1ocnvfvb 7318	. . . . 5 ⊢ ((𝐽:((On × On) × 9o)⟶1-1ontoOn ∧ ⟨(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 1478	. . . 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 2769	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 1570   ∈ wcel 2147  ⟨cotp 4863   × cxp 5656  ^◡ccnv 5657  Oncon0 6388  ⟶1-1ontowf1o 6565  ‘cfv 6566  1^st c1st 8033  2^nd c2nd 8034  9_oc9o 35898  𝐽cj 36024  𝐾₁ck1 36025  𝐾₂ck2 36026  𝐾₃ck3 36027

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-mpt3 7716  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-3o 8516  df-4o 8517  df-oadd 8518  df-omul 8519  df-oi 9541  df-5o 35899  df-6o 35900  df-7o 35901  df-8o 35902  df-9o 35903  df-lexo 36030  df-r0 36031  df-j0 36032  df-j 36033  df-k1 36034  df-k2 36035  df-k3 36036

This theorem is used by:  j0le  36060  k3n0j0lt  36061