Theorem oeicl 6430

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 6424	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
2		1on 6391	. . . 4 ⊢ 1o ∈ On
3	2	a1i 9	. . 3 ⊢ (𝐴 ∈ On → 1o ∈ On)
4		vex 2729	. . . . . . 7 ⊢ 𝑦 ∈ V
5		omcl 6429	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ·o 𝐴) ∈ On)
6		oveq1 5849	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ·o 𝐴) = (𝑦 ·o 𝐴))
7		eqid 2165	. . . . . . . 8 ⊢ (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
8	6, 7	fvmptg 5562	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ (𝑦 ·o 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) = (𝑦 ·o 𝐴))
9	4, 5, 8	sylancr 411	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) = (𝑦 ·o 𝐴))
10	9, 5	eqeltrd 2243	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
11	10	ancoms 266	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
12	11	ralrimiva 2539	. . 3 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘𝑦) ∈ On)
13	3, 12	rdgon 6354	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ On)
14	1, 13	eqeltrd 2243	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ↦ cmpt 4043  Oncon0 4341  ‘cfv 5188  (class class class)co 5842  reccrdg 6337  1_oc1o 6377   ·_o comu 6382   ↑_o coei 6383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-oexpi 6390

This theorem is referenced by: (None)