Theorem oeicl 6515

Description: Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.)

Assertion

Ref	Expression
oeicl	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)

Proof of Theorem oeicl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oeiv 6509	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
2		1on 6476	. . . 4 ⊢ 1o ∈ On
3	2	a1i 9	. . 3 ⊢ (𝐴 ∈ On → 1o ∈ On)
4		vex 2763	. . . . . . 7 ⊢ 𝑦 ∈ V
5		omcl 6514	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ·o 𝐴) ∈ On)
6		oveq1 5925	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ·o 𝐴) = (𝑦 ·o 𝐴))
7		eqid 2193	. . . . . . . 8 ⊢ (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
8	6, 7	fvmptg 5633	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ (𝑦 ·o 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) = (𝑦 ·o 𝐴))
9	4, 5, 8	sylancr 414	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) = (𝑦 ·o 𝐴))
10	9, 5	eqeltrd 2270	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
11	10	ancoms 268	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
12	11	ralrimiva 2567	. . 3 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
13	3, 12	rdgon 6439	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ On)
14	1, 13	eqeltrd 2270	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ↦ cmpt 4090  Oncon0 4394  ‘cfv 5254  (class class class)co 5918  reccrdg 6422  1_oc1o 6462   ·_o comu 6467   ↑_o coei 6468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-omul 6474  df-oexpi 6475

This theorem is referenced by: (None)