Theorem k1on 36056

Description: Values of the 𝐾₁ function are ordinals. (Contributed by BTernaryTau, 11-Oct-2026.)

Assertion

Ref	Expression
k1on	⊢ (𝐾1‘𝐴) ∈ On

Proof of Theorem k1on

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6921	. . . . 5 ⊢ (𝑥 = 𝐴 → (1st ‘(◡𝐽‘𝑥)) = (1st ‘(◡𝐽‘𝐴)))
2	1	fveq2d 6920	. . . 4 ⊢ (𝑥 = 𝐴 → (1st ‘(1st ‘(◡𝐽‘𝑥))) = (1st ‘(1st ‘(◡𝐽‘𝐴))))
3		dfk12 36053	. . . 4 ⊢ 𝐾1 = (𝑥 ∈ On ↦ (1st ‘(1st ‘(◡𝐽‘𝑥))))
4		fvex 6929	. . . 4 ⊢ (1st ‘(1st ‘(◡𝐽‘𝐴))) ∈ V
5	2, 3, 4	fvmpt 7024	. . 3 ⊢ (𝐴 ∈ On → (𝐾1‘𝐴) = (1st ‘(1st ‘(◡𝐽‘𝐴))))
6		jf1o 36052	. . . . 5 ⊢ 𝐽:((On × On) × 9o)⟶1-1ontoOn
7		f1ocnvdm 7324	. . . . 5 ⊢ ((𝐽:((On × On) × 9o)⟶1-1ontoOn ∧ 𝐴 ∈ On) → (◡𝐽‘𝐴) ∈ ((On × On) × 9o))
8	6, 7	mpan 703	. . . 4 ⊢ (𝐴 ∈ On → (◡𝐽‘𝐴) ∈ ((On × On) × 9o))
9		xp1st 8067	. . . 4 ⊢ ((◡𝐽‘𝐴) ∈ ((On × On) × 9o) → (1st ‘(◡𝐽‘𝐴)) ∈ (On × On))
10		xp1st 8067	. . . 4 ⊢ ((1st ‘(◡𝐽‘𝐴)) ∈ (On × On) → (1st ‘(1st ‘(◡𝐽‘𝐴))) ∈ On)
11	8, 9, 10	3syl 19	. . 3 ⊢ (𝐴 ∈ On → (1st ‘(1st ‘(◡𝐽‘𝐴))) ∈ On)
12	5, 11	eqeltrd 2864	. 2 ⊢ (𝐴 ∈ On → (𝐾1‘𝐴) ∈ On)
13	3	fvmptndm 7056	. . 3 ⊢ (¬ 𝐴 ∈ On → (𝐾1‘𝐴) = ∅)
14		0elon 6444	. . 3 ⊢ ∅ ∈ On
15	13, 14	eqeltrdi 2872	. 2 ⊢ (¬ 𝐴 ∈ On → (𝐾1‘𝐴) ∈ On)
16	12, 15	pm2.61i 184	1 ⊢ (𝐾1‘𝐴) ∈ On

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  ∅c0 4281   × cxp 5656  ^◡ccnv 5657  Oncon0 6388  ⟶1-1ontowf1o 6565  ‘cfv 6566  1^st c1st 8033  9_oc9o 35898  𝐽cj 36024  𝐾₁ck1 36025

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

This theorem is used by:  jk1k2k3  36059  j0le  36060  k3n0j0lt  36061  k1le  36062  k2le  36063  k3n0k1lt  36064  k3n0k2lt  36065