Theorem oev 3122

Description: Value of ordinal exponentiation.

Assertion

Ref	Expression
oev	|- ((A e. On /\ B e. 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 e. On
2		difexg 1703	. . . 4 |- (1o e. On -> (1o \ B) e. V)
3	1, 2	ax-mp 6	. . 3 |- (1o \ B) e. V
4		fvex 2838	. . 3 |- (rec({<.x, y>. | y = (x .o A)}, 1o)` B) e. V
5	3, 4	ifex 1797	. 2 |- if(A = (/), (1o \ B), (rec({<.x, y>. | y = (x .o A)}, 1o)` B)) e. 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 e. On /\ v e. On) /\ z = if(w = (/), (1o \ v), (rec({<.x, y>. | y = (x .o w)}, 1o)` v)))}
26	5, 17, 24, 25	oprabval2 3051	1 |- ((A e. On /\ B e. 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   e. wcel 1092  Vcvv 1348   \ cdif 1484  (/)c0 1707  ifcif 1776  {copab 2055  Oncon0 2199  ` cfv 2422  reccrdg 2969  (class class class)co 3001  1oc1o 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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