Theorem oev 8570

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 2744	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2		oveq2 7456	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·o 𝑦) = (𝑥 ·o 𝐴))
3	2	mpteq2dv 5268	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)))
4		rdgeq1 8467	. . . . 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 17	. . . 4 ⊢ (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o) = rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o))
6	5	fveq1d 6922	. . 3 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧))
7	1, 6	ifbieq2d 4574	. 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 4143	. . 3 ⊢ (𝑧 = 𝐵 → (1o ∖ 𝑧) = (1o ∖ 𝐵))
9		fveq2 6920	. . 3 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
10	8, 9	ifeq12d 4569	. 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 8528	. 2 ⊢ ↑o = (𝑦 ∈ On, 𝑧 ∈ On ↦ if(𝑦 = ∅, (1o ∖ 𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧)))
12		1oex 8532	. . . 4 ⊢ 1o ∈ V
13	12	difexi 5348	. . 3 ⊢ (1o ∖ 𝐵) ∈ V
14		fvex 6933	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V
15	13, 14	ifex 4598	. 2 ⊢ if(𝐴 = ∅, (1o ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)) ∈ V
16	7, 10, 11, 15	ovmpo 7610	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 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973  ∅c0 4352  ifcif 4548   ↦ cmpt 5249  Oncon0 6395  ‘cfv 6573  (class class class)co 7448  reccrdg 8465  1_oc1o 8515   ·_o comu 8520   ↑_o coe 8521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-suc 6401  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oexp 8528

This theorem is referenced by:  oevn0  8571  oe0m  8574