Theorem oeicl 6065

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 6059	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
2		vex 2604	. . . . . 6 ⊢ 𝑥 ∈ V
3		omexg 6054	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ On) → (𝑥 ·𝑜 𝐴) ∈ V)
4	2, 3	mpan 414	. . . . 5 ⊢ (𝐴 ∈ On → (𝑥 ·𝑜 𝐴) ∈ V)
5	4	ralrimivw 2435	. . . 4 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ V (𝑥 ·𝑜 𝐴) ∈ V)
6		eqid 2081	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))
7	6	fnmpt 5045	. . . 4 ⊢ (∀𝑥 ∈ V (𝑥 ·𝑜 𝐴) ∈ V → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) Fn V)
8	5, 7	syl 14	. . 3 ⊢ (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) Fn V)
9		1on 6031	. . . 4 ⊢ 1𝑜 ∈ On
10	9	a1i 9	. . 3 ⊢ (𝐴 ∈ On → 1𝑜 ∈ On)
11		omcl 6064	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ·𝑜 𝐴) ∈ On)
12		vex 2604	. . . . . . . 8 ⊢ 𝑦 ∈ V
13		oveq1 5539	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ·𝑜 𝐴) = (𝑦 ·𝑜 𝐴))
14	13, 6	fvmptg 5269	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ (𝑦 ·𝑜 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) = (𝑦 ·𝑜 𝐴))
15	12, 14	mpan 414	. . . . . . 7 ⊢ ((𝑦 ·𝑜 𝐴) ∈ On → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) = (𝑦 ·𝑜 𝐴))
16	11, 15	syl 14	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) = (𝑦 ·𝑜 𝐴))
17	16, 11	eqeltrd 2155	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) ∈ On)
18	17	ancoms 264	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) ∈ On)
19	18	ralrimiva 2434	. . 3 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) ∈ On)
20	8, 10, 19	rdgon 5996	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ On)
21	1, 20	eqeltrd 2155	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  ∀wral 2348  Vcvv 2601   ↦ cmpt 3839  Oncon0 4118   Fn wfn 4917  ‘cfv 4922  (class class class)co 5532  reccrdg 5979  1_𝑜c1o 6017   ·_𝑜 comu 6022   ↑_𝑜 coei 6023

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-oexpi 6030

This theorem is referenced by: (None)