Theorem r1val1 3502

Description: The value of the cumulative hierarchy of sets function expressed recursively. Theorem 7Q of [Enderton] p. 202.

Assertion

Ref	Expression
r1val1	|- (A e. On -> (R1` A) = U.x e. A P~(R1` x))

Distinct variable group(s):   x,A

Proof of Theorem r1val1

Step	Hyp	Ref	Expression
1		onzsl 2367	. . 3 |- (A e. On <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)))
2		0ss 1725	. . . . 5 |- (/) (_ U.x e. A P~(R1` x)
3		fveq2 2832	. . . . . . 7 |- (A = (/) -> (R1` A) = (R1` (/)))
4		r10 3495	. . . . . . 7 |- (R1` (/)) = (/)
5	3, 4	syl6eq 1140	. . . . . 6 |- (A = (/) -> (R1` A) = (/))
6	5	sseq1d 1527	. . . . 5 |- (A = (/) -> ((R1` A) (_ U.x e. A P~(R1` x) <-> (/) (_ U.x e. A P~(R1` x)))
7	2, 6	mpbiri 169	. . . 4 |- (A = (/) -> (R1` A) (_ U.x e. A P~(R1` x))
8		ax-17 925	. . . . . 6 |- (y e. (R1` A) -> A.x y e. (R1` A))
9		hbiu1 2012	. . . . . 6 |- (y e. U.x e. A P~(R1` x) -> A.x y e. U.x e. A P~(R1` x))
10	8, 9	hbss 1501	. . . . 5 |- ((R1` A) (_ U.x e. A P~(R1` x) -> A.x(R1` A) (_ U.x e. A P~(R1` x))
11		fveq2 2832	. . . . . . . 8 |- (A = suc x -> (R1` A) = (R1` suc x))
12		r1suc 3496	. . . . . . . 8 |- (x e. On -> (R1` suc x) = P~(R1` x))
13	11, 12	sylan9eqr 1145	. . . . . . 7 |- ((x e. On /\ A = suc x) -> (R1` A) = P~(R1` x))
14		visset 1350	. . . . . . . . . . 11 |- x e. V
15	14	sucid 2304	. . . . . . . . . 10 |- x e. suc x
16		eleq2 1150	. . . . . . . . . 10 |- (A = suc x -> (x e. A <-> x e. suc x))
17	15, 16	mpbiri 169	. . . . . . . . 9 |- (A = suc x -> x e. A)
18		ssiun2 2019	. . . . . . . . 9 |- (x e. A -> P~(R1` x) (_ U.x e. A P~(R1` x))
19	17, 18	syl 12	. . . . . . . 8 |- (A = suc x -> P~(R1` x) (_ U.x e. A P~(R1` x))
20	19	adantl 305	. . . . . . 7 |- ((x e. On /\ A = suc x) -> P~(R1` x) (_ U.x e. A P~(R1` x))
21	13, 20	eqsstrd 1534	. . . . . 6 |- ((x e. On /\ A = suc x) -> (R1` A) (_ U.x e. A P~(R1` x))
22	21	exp 291	. . . . 5 |- (x e. On -> (A = suc x -> (R1` A) (_ U.x e. A P~(R1` x)))
23	10, 22	r19.23ai 1283	. . . 4 |- (E.x e. On A = suc x -> (R1` A) (_ U.x e. A P~(R1` x))
24		r1lim 3497	. . . . 5 |- ((A e. V /\ Lim A) -> (R1` A) = U.x e. A (R1` x))
25		ordelon 2222	. . . . . . . . . . 11 |- ((Ord A /\ x e. A) -> x e. On)
26		limord 2283	. . . . . . . . . . 11 |- (Lim A -> Ord A)
27	25, 26	sylan 343	. . . . . . . . . 10 |- ((Lim A /\ x e. A) -> x e. On)
28		sucelon 2319	. . . . . . . . . . . 12 |- (x e. On <-> suc x e. On)
29		r1ord2 3500	. . . . . . . . . . . . 13 |- (suc x e. On -> (x e. suc x -> (R1` x) (_ (R1` suc x)))
30	15, 29	mpi 44	. . . . . . . . . . . 12 |- (suc x e. On -> (R1` x) (_ (R1` suc x))
31	28, 30	sylbi 174	. . . . . . . . . . 11 |- (x e. On -> (R1` x) (_ (R1` suc x))
32	31, 12	sseqtrd 1536	. . . . . . . . . 10 |- (x e. On -> (R1` x) (_ P~(R1` x))
33	27, 32	syl 12	. . . . . . . . 9 |- ((Lim A /\ x e. A) -> (R1` x) (_ P~(R1` x))
34	33	exp 291	. . . . . . . 8 |- (Lim A -> (x e. A -> (R1` x) (_ P~(R1` x)))
35	34	r19.21aiv 1259	. . . . . . 7 |- (Lim A -> A.x e. A (R1` x) (_ P~(R1` x))
36		ss2iun 2005	. . . . . . 7 |- (A.x e. A (R1` x) (_ P~(R1` x) -> U.x e. A (R1` x) (_ U.x e. A P~(R1` x))
37	35, 36	syl 12	. . . . . 6 |- (Lim A -> U.x e. A (R1` x) (_ U.x e. A P~(R1` x))
38	37	adantl 305	. . . . 5 |- ((A e. V /\ Lim A) -> U.x e. A (R1` x) (_ U.x e. A P~(R1` x))
39	24, 38	eqsstrd 1534	. . . 4 |- ((A e. V /\ Lim A) -> (R1` A) (_ U.x e. A P~(R1` x))
40	7, 23, 39	3jaoi 633	. . 3 |- ((A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)) -> (R1` A) (_ U.x e. A P~(R1` x))
41	1, 40	sylbi 174	. 2 |- (A e. On -> (R1` A) (_ U.x e. A P~(R1` x))
42		onelon 2223	. . . . . . 7 |- ((A e. On /\ x e. A) -> x e. On)
43	42, 12	syl 12	. . . . . 6 |- ((A e. On /\ x e. A) -> (R1` suc x) = P~(R1` x))
44		r1ord3 3501	. . . . . . 7 |- ((suc x e. On /\ A e. On) -> (suc x (_ A -> (R1` suc x) (_ (R1` A)))
45	42, 28	sylib 173	. . . . . . . 8 |- ((A e. On /\ x e. A) -> suc x e. On)
46		pm3.26 256	. . . . . . . 8 |- ((A e. On /\ x e. A) -> A e. On)
47	45, 46	jca 236	. . . . . . 7 |- ((A e. On /\ x e. A) -> (suc x e. On /\ A e. On))
48		eloni 2209	. . . . . . . . 9 |- (A e. On -> Ord A)
49		ordsucss 2320	. . . . . . . . 9 |- (Ord A -> (x e. A -> suc x (_ A))
50	48, 49	syl 12	. . . . . . . 8 |- (A e. On -> (x e. A -> suc x (_ A))
51	50	imp 277	. . . . . . 7 |- ((A e. On /\ x e. A) -> suc x (_ A)
52	44, 47, 51	sylc 62	. . . . . 6 |- ((A e. On /\ x e. A) -> (R1` suc x) (_ (R1` A))
53	43, 52	eqsstr3d 1535	. . . . 5 |- ((A e. On /\ x e. A) -> P~(R1` x) (_ (R1` A))
54	53	exp 291	. . . 4 |- (A e. On -> (x e. A -> P~(R1` x) (_ (R1` A)))
55	54	r19.21aiv 1259	. . 3 |- (A e. On -> A.x e. A P~(R1` x) (_ (R1` A))
56		iunss 2017	. . 3 |- (U.x e. A P~(R1` x) (_ (R1` A) <-> A.x e. A P~(R1` x) (_ (R1` A))
57	55, 56	sylibr 175	. 2 |- (A e. On -> U.x e. A P~(R1` x) (_ (R1` A))
58	41, 57	eqssd 1518	1 |- (A e. On -> (R1` A) = U.x e. A P~(R1` x))

Syntax hints:   -> wi 2   /\ wa 196   \/ w3o 580   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  Vcvv 1348   (_ wss 1487  (/)c0 1707  P~cpw 1798  U.ciun 1994  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201  ` cfv 2422  R1cr1 3485

This theorem is referenced by:  r1val3 3523

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-rab 1208  df-v 1349  df-sbc 1441  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-iun 1996  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-lim 2204  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-fn 2433  df-fv 2438  df-rdg 2970  df-r1 3487

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