Theorem oev 7481

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 2614	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2		oveq2 6557	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·𝑜 𝑦) = (𝑥 ·𝑜 𝐴))
3	2	mpteq2dv 4673	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)))
4		rdgeq1 7394	. . . . 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 6105	. . 3 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧))
7	1, 6	ifbieq2d 4061	. 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 3684	. . 3 ⊢ (𝑧 = 𝐵 → (1𝑜 ∖ 𝑧) = (1𝑜 ∖ 𝐵))
9		fveq2 6103	. . 3 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
10	8, 9	ifeq12d 4056	. 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 7453	. 2 ⊢ ↑𝑜 = (𝑦 ∈ On, 𝑧 ∈ On ↦ if(𝑦 = ∅, (1𝑜 ∖ 𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝑦)), 1𝑜)‘𝑧)))
12		1on 7454	. . . . 5 ⊢ 1𝑜 ∈ On
13	12	elexi 3186	. . . 4 ⊢ 1𝑜 ∈ V
14		difss 3699	. . . 4 ⊢ (1𝑜 ∖ 𝐵) ⊆ 1𝑜
15	13, 14	ssexi 4731	. . 3 ⊢ (1𝑜 ∖ 𝐵) ∈ V
16		fvex 6113	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∈ V
17	15, 16	ifex 4106	. 2 ⊢ if(𝐴 = ∅, (1𝑜 ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)) ∈ V
18	7, 10, 11, 17	ovmpt2 6694	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 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  ifcif 4036   ↦ cmpt 4643  Oncon0 5640  ‘cfv 5804  (class class class)co 6549  reccrdg 7392  1_𝑜c1o 7440   ·_𝑜 comu 7445   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oexp 7453

This theorem is referenced by:  oevn0  7482  oe0m  7485