Theorem oeiv 6415

Description: Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.)

Assertion

Ref	Expression
oeiv	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oeiv

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

Step	Hyp	Ref	Expression
1		1on 6382	. . 3 ⊢ 1o ∈ On
2		vex 2724	. . . . . . 7 ⊢ 𝑥 ∈ V
3		omexg 6410	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ On) → (𝑥 ·o 𝐴) ∈ V)
4	2, 3	mpan 421	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑥 ·o 𝐴) ∈ V)
5	4	ralrimivw 2538	. . . . 5 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ V (𝑥 ·o 𝐴) ∈ V)
6		eqid 2164	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
7	6	fnmpt 5308	. . . . 5 ⊢ (∀𝑥 ∈ V (𝑥 ·o 𝐴) ∈ V → (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) Fn V)
8	5, 7	syl 14	. . . 4 ⊢ (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) Fn V)
9		rdgexggg 6336	. . . 4 ⊢ (((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) Fn V ∧ 1o ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V)
10	8, 9	syl3an1 1260	. . 3 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V)
11	1, 10	mp3an2 1314	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V)
12		oveq2 5844	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·o 𝑦) = (𝑥 ·o 𝐴))
13	12	mpteq2dv 4067	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)))
14		rdgeq1 6330	. . . . 5 ⊢ ((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) → rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o) = rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o))
15	13, 14	syl 14	. . . 4 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o) = rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o))
16	15	fveq1d 5482	. . 3 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧))
17		fveq2 5480	. . 3 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
18		df-oexpi 6381	. . 3 ⊢ ↑o = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧))
19	16, 17, 18	ovmpog 5967	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
20	11, 19	mpd3an3 1327	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1342   ∈ wcel 2135  ∀wral 2442  Vcvv 2721   ↦ cmpt 4037  Oncon0 4335   Fn wfn 5177  ‘cfv 5182  (class class class)co 5836  reccrdg 6328  1_oc1o 6368   ·_o comu 6373   ↑_o coei 6374

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-iord 4338  df-on 4340  df-suc 4343  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-1o 6375  df-oadd 6379  df-omul 6380  df-oexpi 6381

This theorem is referenced by:  oei0  6418  oeicl  6421