Theorem oev 8499

Description: Value of ordinal exponentiation. (Contributed by NM, 30-Dec-2004.) (Revised by Mario Carneiro, 23-Aug-2014.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oev

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

Step	Hyp	Ref	Expression
1		eqeq1 2773	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2		oveq2 7419	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·o 𝑦) = (𝑥 ·o 𝐴))
3	2	mpteq2dv 5209	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)))
4		rdgeq1 8398	. . . . 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))
5	3, 4	syl 18	. . . 4 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o) = rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o))
6	5	fveq1d 6884	. . 3 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧))
7	1, 6	ifbieq2d 4519	. 2 ⊢ (𝑦 = 𝐴 → if(𝑦 = ∅, (1o ∖ 𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧)) = if(𝐴 = ∅, (1o ∖ 𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧)))
8		difeq2 4083	. . 3 ⊢ (𝑧 = 𝐵 → (1o ∖ 𝑧) = (1o ∖ 𝐵))
9		fveq2 6882	. . 3 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
10	8, 9	ifeq12d 4514	. 2 ⊢ (𝑧 = 𝐵 → if(𝐴 = ∅, (1o ∖ 𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧)) = if(𝐴 = ∅, (1o ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
11		df-oexp 8459	. 2 ⊢ ↑o = (𝑦 ∈ On, 𝑧 ∈ On ↦ if(𝑦 = ∅, (1o ∖ 𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧)))
12		1oex 8463	. . . 4 ⊢ 1o ∈ V
13	12	difexi 5301	. . 3 ⊢ (1o ∖ 𝐵) ∈ V
14		fvex 6895	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V
15	13, 14	ifex 4543	. 2 ⊢ if(𝐴 = ∅, (1o ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)) ∈ V
16	7, 10, 11, 15	ovmpo 7571	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = if(𝐴 = ∅, (1o ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910  ∅c0 4294  ifcif 4492   ↦ cmpt 5196  Oncon0 6361  ‘cfv 6537  (class class class)co 7411  reccrdg 8396  1_oc1o 8446   ·_o comu 8451   ↑_o coe 8452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-suc 6367  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oexp 8459

This theorem is referenced by:  oevn0  8500  oe0m  8503