Theorem oeicl 6042

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 6036	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
2		vex 2560	. . . . . 6 ⊢ 𝑥 ∈ V
3		omexg 6031	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ On) → (𝑥 ·𝑜 𝐴) ∈ V)
4	2, 3	mpan 400	. . . . 5 ⊢ (𝐴 ∈ On → (𝑥 ·𝑜 𝐴) ∈ V)
5	4	ralrimivw 2393	. . . 4 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ V (𝑥 ·𝑜 𝐴) ∈ V)
6		eqid 2040	. . . . 5 ⊢ (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))
7	6	fnmpt 5025	. . . 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 6008	. . . 4 ⊢ 1𝑜 ∈ On
10	9	a1i 9	. . 3 ⊢ (𝐴 ∈ On → 1𝑜 ∈ On)
11		omcl 6041	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ·𝑜 𝐴) ∈ On)
12		vex 2560	. . . . . . . 8 ⊢ 𝑦 ∈ V
13		oveq1 5519	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ·𝑜 𝐴) = (𝑦 ·𝑜 𝐴))
14	13, 6	fvmptg 5248	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ (𝑦 ·𝑜 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) = (𝑦 ·𝑜 𝐴))
15	12, 14	mpan 400	. . . . . . 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 2114	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) ∈ On)
18	17	ancoms 255	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) ∈ On)
19	18	ralrimiva 2392	. . 3 ⊢ (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘𝑦) ∈ On)
20	8, 10, 19	rdgon 5973	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ On)
21	1, 20	eqeltrd 2114	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∀wral 2306  Vcvv 2557   ↦ cmpt 3818  Oncon0 4100   Fn wfn 4897  ‘cfv 4902  (class class class)co 5512  reccrdg 5956  1_𝑜c1o 5994   ·_𝑜 comu 5999   ↑_𝑜 coei 6000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-oexpi 6007

This theorem is referenced by: (None)