Theorem r1val1 4658

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 3117	. . 3 |- (A e. On <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)))
2		0ss 2301	. . . . 5 |- (/) (_ U_x e. A P~(R1` x)
3		fveq2 3724	. . . . . . 7 |- (A = (/) -> (R1` A) = (R1` (/)))
4		r10 4651	. . . . . . 7 |- (R1` (/)) = (/)
5	3, 4	syl6eq 1523	. . . . . 6 |- (A = (/) -> (R1` A) = (/))
6	5	sseq1d 2088	. . . . 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-17 971	. . . . . 6 |- (y e. (R1` A) -> A.x y e. (R1` A))
9		hbiu1 2584	. . . . . 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 2062	. . . . 5 |- ((R1` A) (_ U_x e. A P~(R1` x) -> A.x(R1` A) (_ U_x e. A P~(R1` x))
11		fveq2 3724	. . . . . . . 8 |- (A = suc x -> (R1` A) = (R1` suc x))
12		r1suc 4652	. . . . . . . 8 |- (x e. On -> (R1` suc x) = P~(R1` x))
13	11, 12	sylan9eqr 1529	. . . . . . 7 |- ((x e. On /\ A = suc x) -> (R1` A) = P~(R1` x))
14		visset 1813	. . . . . . . . . . 11 |- x e. V
15	14	sucid 3051	. . . . . . . . . 10 |- x e. suc x
16		eleq2 1535	. . . . . . . . . 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 2593	. . . . . . . . 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 388	. . . . . . 7 |- ((x e. On /\ A = suc x) -> P~(R1` x) (_ U_x e. A P~(R1` x))
21	13, 20	eqsstrd 2095	. . . . . 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 1742	. . . 4 |- (E.x e. On A = suc x -> (R1` A) (_ U_x e. A P~(R1` x))
24		r1lim 4653	. . . . 5 |- ((A e. V /\ Lim A) -> (R1` A) = U_x e. A (R1` x))
25		ordelon 2971	. . . . . . . . . 10 |- ((Ord A /\ x e. A) -> x e. On)
26		limord 3028	. . . . . . . . . 10 |- (Lim A -> Ord A)
27	25, 26	sylan 448	. . . . . . . . 9 |- ((Lim A /\ x e. A) -> x e. On)
28		sucelon 3068	. . . . . . . . . . 11 |- (x e. On <-> suc x e. On)
29		r1ord2 4656	. . . . . . . . . . . 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 2097	. . . . . . . . 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 1714	. . . . . . 7 |- (Lim A -> A.x e. A (R1` x) (_ P~(R1` x))
35		ss2iun 2577	. . . . . . 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 388	. . . . 5 |- ((A e. V /\ Lim A) -> U_x e. A (R1` x) (_ U_x e. A P~(R1` x))
38	24, 37	eqsstrd 2095	. . . 4 |- ((A e. V /\ Lim A) -> (R1` A) (_ U_x e. A P~(R1` x))
39	7, 23, 38	3jaoi 887	. . 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 2972	. . . . . 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 4657	. . . . . 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 2958	. . . . . . . 8 |- (A e. On -> Ord A)
48		ordsucss 3069	. . . . . . . 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 2096	. . . 4 |- ((A e. On /\ x e. A) -> P~(R1` x) (_ (R1` A))
53	52	r19.21aiva 1714	. . 3 |- (A e. On -> A.x e. A P~(R1` x) (_ (R1` A))
54		iunss 2591	. . 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 2079	1 |- (A e. On -> (R1` A) = U_x e. A P~(R1` x))

Syntax hints:   -> wi 3   /\ wa 223   \/ w3o 774   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  Vcvv 1811   (_ wss 2047  (/)c0 2280  P~cpw 2401  U_ciun 2566  Ord word 2947  Oncon0 2948  Lim wlim 2949  suc csuc 2950  ` cfv 3182  R1cr1 4641

This theorem is referenced by:  r1val3 4679

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-r1 4643

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