Theorem r1val1 4668

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:   x,A

Proof of Theorem r1val1

Step	Hyp	Ref	Expression
1		onzsl 3123	. . 3 |- (A e. On <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)))
2		0ss 2305	. . . . 5 |- (/) (_ U_x e. A P~(R1` x)
3		fveq2 3730	. . . . . . 7 |- (A = (/) -> (R1` A) = (R1` (/)))
4		r10 4661	. . . . . . 7 |- (R1` (/)) = (/)
5	3, 4	syl6eq 1526	. . . . . 6 |- (A = (/) -> (R1` A) = (/))
6	5	sseq1d 2091	. . . . 5 |- (A = (/) -> ((R1` A) (_ U_x e. A P~(R1` x) <-> (/) (_ U_x e. A P~(R1` x)))
7	2, 6	mpbiri 194	. . . 4 |- (A = (/) -> (R1` A) (_ U_x e. A P~(R1` x))
8		ax-17 973	. . . . . 6 |- (y e. (R1` A) -> A.x y e. (R1` A))
9		hbiu1 2588	. . . . . 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 2065	. . . . 5 |- ((R1` A) (_ U_x e. A P~(R1` x) -> A.x(R1` A) (_ U_x e. A P~(R1` x))
11		fveq2 3730	. . . . . . . 8 |- (A = suc x -> (R1` A) = (R1` suc x))
12		r1suc 4662	. . . . . . . 8 |- (x e. On -> (R1` suc x) = P~(R1` x))
13	11, 12	sylan9eqr 1532	. . . . . . 7 |- ((x e. On /\ A = suc x) -> (R1` A) = P~(R1` x))
14		visset 1816	. . . . . . . . . . 11 |- x e. V
15	14	sucid 3057	. . . . . . . . . 10 |- x e. suc x
16		eleq2 1538	. . . . . . . . . 10 |- (A = suc x -> (x e. A <-> x e. suc x))
17	15, 16	mpbiri 194	. . . . . . . . 9 |- (A = suc x -> x e. A)
18		ssiun2 2597	. . . . . . . . 9 |- (x e. A -> P~(R1` x) (_ U_x e. A P~(R1` x))
19	17, 18	syl 10	. . . . . . . 8 |- (A = suc x -> P~(R1` x) (_ U_x e. A P~(R1` x))
20	19	adantl 390	. . . . . . 7 |- ((x e. On /\ A = suc x) -> P~(R1` x) (_ U_x e. A P~(R1` x))
21	13, 20	eqsstrd 2098	. . . . . 6 |- ((x e. On /\ A = suc x) -> (R1` A) (_ U_x e. A P~(R1` x))
22	21	ex 373	. . . . 5 |- (x e. On -> (A = suc x -> (R1` A) (_ U_x e. A P~(R1` x)))
23	10, 22	r19.23ai 1745	. . . 4 |- (E.x e. On A = suc x -> (R1` A) (_ U_x e. A P~(R1` x))
24		r1lim 4663	. . . . 5 |- ((A e. V /\ Lim A) -> (R1` A) = U_x e. A (R1` x))
25		ordelon 2977	. . . . . . . . . 10 |- ((Ord A /\ x e. A) -> x e. On)
26		limord 3034	. . . . . . . . . 10 |- (Lim A -> Ord A)
27	25, 26	sylan 450	. . . . . . . . 9 |- ((Lim A /\ x e. A) -> x e. On)
28		sucelon 3074	. . . . . . . . . . 11 |- (x e. On <-> suc x e. On)
29		r1ord2 4666	. . . . . . . . . . . 12 |- (suc x e. On -> (x e. suc x -> (R1` x) (_ (R1` suc x)))
30	15, 29	mpi 44	. . . . . . . . . . 11 |- (suc x e. On -> (R1` x) (_ (R1` suc x))
31	28, 30	sylbi 199	. . . . . . . . . 10 |- (x e. On -> (R1` x) (_ (R1` suc x))
32	31, 12	sseqtrd 2100	. . . . . . . . 9 |- (x e. On -> (R1` x) (_ P~(R1` x))
33	27, 32	syl 10	. . . . . . . 8 |- ((Lim A /\ x e. A) -> (R1` x) (_ P~(R1` x))
34	33	r19.21aiva 1717	. . . . . . 7 |- (Lim A -> A.x e. A (R1` x) (_ P~(R1` x))
35		ss2iun 2581	. . . . . . 7 |- (A.x e. A (R1` x) (_ P~(R1` x) -> U_x e. A (R1` x) (_ U_x e. A P~(R1` x))
36	34, 35	syl 10	. . . . . 6 |- (Lim A -> U_x e. A (R1` x) (_ U_x e. A P~(R1` x))
37	36	adantl 390	. . . . 5 |- ((A e. V /\ Lim A) -> U_x e. A (R1` x) (_ U_x e. A P~(R1` x))
38	24, 37	eqsstrd 2098	. . . 4 |- ((A e. V /\ Lim A) -> (R1` A) (_ U_x e. A P~(R1` x))
39	7, 23, 38	3jaoi 889	. . 3 |- ((A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)) -> (R1` A) (_ U_x e. A P~(R1` x))
40	1, 39	sylbi 199	. 2 |- (A e. On -> (R1` A) (_ U_x e. A P~(R1` x))
41		onelon 2978	. . . . . 6 |- ((A e. On /\ x e. A) -> x e. On)
42	41, 12	syl 10	. . . . 5 |- ((A e. On /\ x e. A) -> (R1` suc x) = P~(R1` x))
43		r1ord3 4667	. . . . . 6 |- ((suc x e. On /\ A e. On) -> (suc x (_ A -> (R1` suc x) (_ (R1` A)))
44	41, 28	sylib 198	. . . . . . 7 |- ((A e. On /\ x e. A) -> suc x e. On)
45		pm3.26 319	. . . . . . 7 |- ((A e. On /\ x e. A) -> A e. On)
46	44, 45	jca 288	. . . . . 6 |- ((A e. On /\ x e. A) -> (suc x e. On /\ A e. On))
47		eloni 2964	. . . . . . . 8 |- (A e. On -> Ord A)
48		ordsucss 3075	. . . . . . . 8 |- (Ord A -> (x e. A -> suc x (_ A))
49	47, 48	syl 10	. . . . . . 7 |- (A e. On -> (x e. A -> suc x (_ A))
50	49	imp 350	. . . . . 6 |- ((A e. On /\ x e. A) -> suc x (_ A)
51	43, 46, 50	sylc 68	. . . . 5 |- ((A e. On /\ x e. A) -> (R1` suc x) (_ (R1` A))
52	42, 51	eqsstr3d 2099	. . . 4 |- ((A e. On /\ x e. A) -> P~(R1` x) (_ (R1` A))
53	52	r19.21aiva 1717	. . 3 |- (A e. On -> A.x e. A P~(R1` x) (_ (R1` A))
54		iunss 2595	. . 3 |- (U_x e. A P~(R1` x) (_ (R1` A) <-> A.x e. A P~(R1` x) (_ (R1` A))
55	53, 54	sylibr 200	. 2 |- (A e. On -> U_x e. A P~(R1` x) (_ (R1` A))
56	40, 55	eqssd 2082	1 |- (A e. On -> (R1` A) = U_x e. A P~(R1` x))

Syntax hints:   -> wi 3   /\ wa 223   \/ w3o 776   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649  Vcvv 1814   (_ wss 2050  (/)c0 2283  P~cpw 2405  U_ciun 2570  Ord word 2953  Oncon0 2954  Lim wlim 2955  suc csuc 2956  ` cfv 3188  R1cr1 4651

This theorem is referenced by:  r1val3 4689

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-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-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-r1 4653

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