Theorem oev 3122

Description: Value of ordinal exponentiation.

Assertion

Ref	Expression
oev	⊢ ((A ∈ On ∧ B ∈ On) → (A ↑o B) = if(A = ∅, (1o ∖ B), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B)))

Distinct variable group(s):   x,y,A

Proof of Theorem oev

Step	Hyp	Ref	Expression
1		1o 3109	. . . 4 ⊢ 1o ∈ On
2		difexg 1703	. . . 4 ⊢ (1o ∈ On → (1o ∖ B) ∈ V)
3	1, 2	ax-mp 6	. . 3 ⊢ (1o ∖ B) ∈ V
4		fvex 2838	. . 3 ⊢ (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∈ V
5	3, 4	ifex 1797	. 2 ⊢ if(A = ∅, (1o ∖ B), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B)) ∈ V
6		cleq1 1107	. . . 4 ⊢ (w = A → (w = ∅ ↔ A = ∅))
7		ifbi 1783	. . . 4 ⊢ ((w = ∅ ↔ A = ∅) → if(w = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o w)}, 1o) ‘v)) = if(A = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o w)}, 1o) ‘v)))
8	6, 7	syl 12	. . 3 ⊢ (w = A → if(w = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o w)}, 1o) ‘v)) = if(A = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o w)}, 1o) ‘v)))
9		opreq2 3007	. . . . . . 7 ⊢ (w = A → (x ·o w) = (x ·o A))
10	9	cleq2d 1112	. . . . . 6 ⊢ (w = A → (y = (x ·o w) ↔ y = (x ·o A)))
11	10	biopabdv 2102	. . . . 5 ⊢ (w = A → {⟨x, y⟩∣y = (x ·o w)} = {⟨x, y⟩∣y = (x ·o A)})
12		rdgeq1 2972	. . . . 5 ⊢ ({⟨x, y⟩∣y = (x ·o w)} = {⟨x, y⟩∣y = (x ·o A)} → rec({⟨x, y⟩∣y = (x ·o w)}, 1o) = rec({⟨x, y⟩∣y = (x ·o A)}, 1o))
13		fveq1 2831	. . . . 5 ⊢ (rec({⟨x, y⟩∣y = (x ·o w)}, 1o) = rec({⟨x, y⟩∣y = (x ·o A)}, 1o) → (rec({⟨x, y⟩∣y = (x ·o w)}, 1o) ‘v) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v))
14	11, 12, 13	3syl 21	. . . 4 ⊢ (w = A → (rec({⟨x, y⟩∣y = (x ·o w)}, 1o) ‘v) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v))
15		ifeq2 1779	. . . 4 ⊢ ((rec({⟨x, y⟩∣y = (x ·o w)}, 1o) ‘v) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v) → if(A = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o w)}, 1o) ‘v)) = if(A = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v)))
16	14, 15	syl 12	. . 3 ⊢ (w = A → if(A = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o w)}, 1o) ‘v)) = if(A = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v)))
17	8, 16	eqtrd 1128	. 2 ⊢ (w = A → if(w = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o w)}, 1o) ‘v)) = if(A = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v)))
18		difeq2 1583	. . . 4 ⊢ (v = B → (1o ∖ v) = (1o ∖ B))
19		ifeq1 1778	. . . 4 ⊢ ((1o ∖ v) = (1o ∖ B) → if(A = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v)) = if(A = ∅, (1o ∖ B), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v)))
20	18, 19	syl 12	. . 3 ⊢ (v = B → if(A = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v)) = if(A = ∅, (1o ∖ B), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v)))
21		fveq2 2832	. . . 4 ⊢ (v = B → (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B))
22		ifeq2 1779	. . . 4 ⊢ ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) → if(A = ∅, (1o ∖ B), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v)) = if(A = ∅, (1o ∖ B), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B)))
23	21, 22	syl 12	. . 3 ⊢ (v = B → if(A = ∅, (1o ∖ B), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v)) = if(A = ∅, (1o ∖ B), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B)))
24	20, 23	eqtrd 1128	. 2 ⊢ (v = B → if(A = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘v)) = if(A = ∅, (1o ∖ B), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B)))
25		df-oexp 3108	. 2 ⊢ ↑o = {⟨⟨w, v⟩, z⟩∣((w ∈ On ∧ v ∈ On) ∧ z = if(w = ∅, (1o ∖ v), (rec({⟨x, y⟩∣y = (x ·o w)}, 1o) ‘v)))}
26	5, 17, 24, 25	oprabval2 3051	1 ⊢ ((A ∈ On ∧ B ∈ On) → (A ↑o B) = if(A = ∅, (1o ∖ B), (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484  ∅c0 1707  ifcif 1776  {copab 2055  Oncon0 2199   ‘cfv 2422  reccrdg 2969  (class class class)co 3001  1_oc1o 3099   ·_o comu 3102   ↑_o coe 3103

This theorem is referenced by:  oevn0 3123  oe0m 3127

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oexp 3108

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