Theorem oev2 4168

Description: Alternate value of ordinal exponentiation. Compare oev 4159.

Assertion

Ref	Expression
oev2	⊢ ((A ∈ On ⋀ B ∈ On) → (A ↑o B) = ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ((V ∖ ∩A) ∪ ∩B)))

Distinct variable group:   x,y,A

Proof of Theorem oev2

Step	Hyp	Ref	Expression
1		oe0m0 4165	. . . . 5 ⊢ (∅ ↑o ∅) = 1o
2		opreq12 3976	. . . . 5 ⊢ ((A = ∅ ⋀ B = ∅) → (A ↑o B) = (∅ ↑o ∅))
3		fveq2 3730	. . . . . . . 8 ⊢ (B = ∅ → (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘∅))
4		1on 4144	. . . . . . . . . 10 ⊢ 1o ∈ On
5	4	elisseti 1821	. . . . . . . . 9 ⊢ 1o ∈ V
6	5	rdg0 3947	. . . . . . . 8 ⊢ (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘∅) = 1o
7	3, 6	syl6eq 1526	. . . . . . 7 ⊢ (B = ∅ → (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) = 1o)
8		inteq 2540	. . . . . . . 8 ⊢ (B = ∅ → ∩B = ∩∅)
9		int0 2551	. . . . . . . 8 ⊢ ∩∅ = V
10	8, 9	syl6eq 1526	. . . . . . 7 ⊢ (B = ∅ → ∩B = V)
11	7, 10	ineq12d 2221	. . . . . 6 ⊢ (B = ∅ → ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ∩B) = (1o ∩ V))
12		inv1 2303	. . . . . . 7 ⊢ (1o ∩ V) = 1o
13	12	a1i 8	. . . . . 6 ⊢ (A = ∅ → (1o ∩ V) = 1o)
14	11, 13	sylan9eqr 1532	. . . . 5 ⊢ ((A = ∅ ⋀ B = ∅) → ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ∩B) = 1o)
15	1, 2, 14	3eqtr4a 1535	. . . 4 ⊢ ((A = ∅ ⋀ B = ∅) → (A ↑o B) = ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ∩B))
16		opreq1 3974	. . . . . . 7 ⊢ (A = ∅ → (A ↑o B) = (∅ ↑o B))
17		oe0m1 4166	. . . . . . . 8 ⊢ (B ∈ On → (∅ ∈ B ↔ (∅ ↑o B) = ∅))
18	17	biimpa 418	. . . . . . 7 ⊢ ((B ∈ On ⋀ ∅ ∈ B) → (∅ ↑o B) = ∅)
19	16, 18	sylan9eqr 1532	. . . . . 6 ⊢ (((B ∈ On ⋀ ∅ ∈ B) ⋀ A = ∅) → (A ↑o B) = ∅)
20	19	an1rs 491	. . . . 5 ⊢ (((B ∈ On ⋀ A = ∅) ⋀ ∅ ∈ B) → (A ↑o B) = ∅)
21		int0el 2565	. . . . . . . 8 ⊢ (∅ ∈ B → ∩B = ∅)
22	21	ineq2d 2220	. . . . . . 7 ⊢ (∅ ∈ B → ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ∩B) = ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ∅))
23		in0 2302	. . . . . . 7 ⊢ ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ∅) = ∅
24	22, 23	syl6eq 1526	. . . . . 6 ⊢ (∅ ∈ B → ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ∩B) = ∅)
25	24	adantl 390	. . . . 5 ⊢ (((B ∈ On ⋀ A = ∅) ⋀ ∅ ∈ B) → ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ∩B) = ∅)
26	20, 25	eqtr4d 1513	. . . 4 ⊢ (((B ∈ On ⋀ A = ∅) ⋀ ∅ ∈ B) → (A ↑o B) = ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ∩B))
27	15, 26	oe0lem 4158	. . 3 ⊢ ((B ∈ On ⋀ A = ∅) → (A ↑o B) = ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ∩B))
28		inteq 2540	. . . . . . . . . 10 ⊢ (A = ∅ → ∩A = ∩∅)
29	28, 9	syl6eq 1526	. . . . . . . . 9 ⊢ (A = ∅ → ∩A = V)
30	29	difeq2d 2162	. . . . . . . 8 ⊢ (A = ∅ → (V ∖ ∩A) = (V ∖ V))
31		difid 2338	. . . . . . . 8 ⊢ (V ∖ V) = ∅
32	30, 31	syl6eq 1526	. . . . . . 7 ⊢ (A = ∅ → (V ∖ ∩A) = ∅)
33	32	uneq2d 2187	. . . . . 6 ⊢ (A = ∅ → (∩B ∪ (V ∖ ∩A)) = (∩B ∪ ∅))
34		uncom 2179	. . . . . 6 ⊢ (∩B ∪ (V ∖ ∩A)) = ((V ∖ ∩A) ∪ ∩B)
35		un0 2301	. . . . . 6 ⊢ (∩B ∪ ∅) = ∩B
36	33, 34, 35	3eqtr3g 1533	. . . . 5 ⊢ (A = ∅ → ((V ∖ ∩A) ∪ ∩B) = ∩B)
37	36	adantl 390	. . . 4 ⊢ ((B ∈ On ⋀ A = ∅) → ((V ∖ ∩A) ∪ ∩B) = ∩B)
38	37	ineq2d 2220	. . 3 ⊢ ((B ∈ On ⋀ A = ∅) → ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ((V ∖ ∩A) ∪ ∩B)) = ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ∩B))
39	27, 38	eqtr4d 1513	. 2 ⊢ ((B ∈ On ⋀ A = ∅) → (A ↑o B) = ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ((V ∖ ∩A) ∪ ∩B)))
40		oevn0 4160	. . 3 ⊢ (((A ∈ On ⋀ B ∈ On) ⋀ ∅ ∈ A) → (A ↑o B) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B))
41		int0el 2565	. . . . . . . . . 10 ⊢ (∅ ∈ A → ∩A = ∅)
42	41	difeq2d 2162	. . . . . . . . 9 ⊢ (∅ ∈ A → (V ∖ ∩A) = (V ∖ ∅))
43		dif0 2339	. . . . . . . . 9 ⊢ (V ∖ ∅) = V
44	42, 43	syl6eq 1526	. . . . . . . 8 ⊢ (∅ ∈ A → (V ∖ ∩A) = V)
45	44	uneq2d 2187	. . . . . . 7 ⊢ (∅ ∈ A → (∩B ∪ (V ∖ ∩A)) = (∩B ∪ V))
46		unv 2304	. . . . . . 7 ⊢ (∩B ∪ V) = V
47	45, 34, 46	3eqtr3g 1533	. . . . . 6 ⊢ (∅ ∈ A → ((V ∖ ∩A) ∪ ∩B) = V)
48	47	adantl 390	. . . . 5 ⊢ (((A ∈ On ⋀ B ∈ On) ⋀ ∅ ∈ A) → ((V ∖ ∩A) ∪ ∩B) = V)
49	48	ineq2d 2220	. . . 4 ⊢ (((A ∈ On ⋀ B ∈ On) ⋀ ∅ ∈ A) → ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ((V ∖ ∩A) ∪ ∩B)) = ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ V))
50		inv1 2303	. . . 4 ⊢ ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ V) = (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B)
51	49, 50	syl6req 1527	. . 3 ⊢ (((A ∈ On ⋀ B ∈ On) ⋀ ∅ ∈ A) → (rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) = ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ((V ∖ ∩A) ∪ ∩B)))
52	40, 51	eqtrd 1510	. 2 ⊢ (((A ∈ On ⋀ B ∈ On) ⋀ ∅ ∈ A) → (A ↑o B) = ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ((V ∖ ∩A) ∪ ∩B)))
53	39, 52	oe0lem 4158	1 ⊢ ((A ∈ On ⋀ B ∈ On) → (A ↑o B) = ((rec({⟨x, y⟩∣y = (x ·o A)}, 1o) ‘B) ∩ ((V ∖ ∩A) ∪ ∩B)))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  Vcvv 1814   ∖ cdif 2047   ∪ cun 2048   ∩ cin 2049  ∅c0 2283  ∩cint 2537  {copab 2671  Oncon0 2954   ‘cfv 3188  reccrdg 3937  (class class class)co 3969  1_oc1o 4134   ·_o comu 4137   ↑_o coe 4138

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1o 4139  df-oexp 4143

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