Theorem oev 8429

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 2735	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
2		oveq2 7354	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ·o 𝑦) = (𝑥 ·o 𝐴))
3	2	mpteq2dv 5185	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 ·o 𝑦)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)))
4		rdgeq1 8330	. . . . 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 6824	. . 3 ⊢ (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧))
7	1, 6	ifbieq2d 4502	. 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 4070	. . 3 ⊢ (𝑧 = 𝐵 → (1o ∖ 𝑧) = (1o ∖ 𝐵))
9		fveq2 6822	. . 3 ⊢ (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
10	8, 9	ifeq12d 4497	. 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 8391	. 2 ⊢ ↑o = (𝑦 ∈ On, 𝑧 ∈ On ↦ if(𝑦 = ∅, (1o ∖ 𝑧), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝑦)), 1o)‘𝑧)))
12		1oex 8395	. . . 4 ⊢ 1o ∈ V
13	12	difexi 5268	. . 3 ⊢ (1o ∖ 𝐵) ∈ V
14		fvex 6835	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵) ∈ V
15	13, 14	ifex 4526	. 2 ⊢ if(𝐴 = ∅, (1o ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)) ∈ V
16	7, 10, 11, 15	ovmpo 7506	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 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3899  ∅c0 4283  ifcif 4475   ↦ cmpt 5172  Oncon0 6306  ‘cfv 6481  (class class class)co 7346  reccrdg 8328  1_oc1o 8378   ·_o comu 8383   ↑_o coe 8384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-suc 6312  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oexp 8391

This theorem is referenced by:  oevn0  8430  oe0m  8433