Theorem r1val1 4630

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 3107	. . 3 |- (A e. On <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)))
2		0ss 2291	. . . . 5 |- (/) (_ U_x e. A P~(R1` x)
3		fveq2 3709	. . . . . . 7 |- (A = (/) -> (R1` A) = (R1` (/)))
4		r10 4623	. . . . . . 7 |- (R1` (/)) = (/)
5	3, 4	syl6eq 1515	. . . . . 6 |- (A = (/) -> (R1` A) = (/))
6	5	sseq1d 2078	. . . . 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-17 968	. . . . . 6 |- (y e. (R1` A) -> A.x y e. (R1` A))
9		hbiu1 2574	. . . . . 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 2052	. . . . 5 |- ((R1` A) (_ U_x e. A P~(R1` x) -> A.x(R1` A) (_ U_x e. A P~(R1` x))
11		fveq2 3709	. . . . . . . 8 |- (A = suc x -> (R1` A) = (R1` suc x))
12		r1suc 4624	. . . . . . . 8 |- (x e. On -> (R1` suc x) = P~(R1` x))
13	11, 12	sylan9eqr 1521	. . . . . . 7 |- ((x e. On /\ A = suc x) -> (R1` A) = P~(R1` x))
14		visset 1804	. . . . . . . . . . 11 |- x e. V
15	14	sucid 3041	. . . . . . . . . 10 |- x e. suc x
16		eleq2 1527	. . . . . . . . . 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 2583	. . . . . . . . 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 2085	. . . . . 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 1734	. . . 4 |- (E.x e. On A = suc x -> (R1` A) (_ U_x e. A P~(R1` x))
24		r1lim 4625	. . . . 5 |- ((A e. V /\ Lim A) -> (R1` A) = U_x e. A (R1` x))
25		ordelon 2961	. . . . . . . . . 10 |- ((Ord A /\ x e. A) -> x e. On)
26		limord 3018	. . . . . . . . . 10 |- (Lim A -> Ord A)
27	25, 26	sylan 448	. . . . . . . . 9 |- ((Lim A /\ x e. A) -> x e. On)
28		sucelon 3058	. . . . . . . . . . 11 |- (x e. On <-> suc x e. On)
29		r1ord2 4628	. . . . . . . . . . . 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 2087	. . . . . . . . 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 1706	. . . . . . 7 |- (Lim A -> A.x e. A (R1` x) (_ P~(R1` x))
35		ss2iun 2567	. . . . . . 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 2085	. . . 4 |- ((A e. V /\ Lim A) -> (R1` A) (_ U_x e. A P~(R1` x))
39	7, 23, 38	3jaoi 884	. . 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 2962	. . . . . 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 4629	. . . . . 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 2948	. . . . . . . 8 |- (A e. On -> Ord A)
48		ordsucss 3059	. . . . . . . 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 2086	. . . 4 |- ((A e. On /\ x e. A) -> P~(R1` x) (_ (R1` A))
53	52	r19.21aiva 1706	. . 3 |- (A e. On -> A.x e. A P~(R1` x) (_ (R1` A))
54		iunss 2581	. . 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 2069	1 |- (A e. On -> (R1` A) = U_x e. A P~(R1` x))

Syntax hints:   -> wi 3   /\ wa 223   \/ w3o 772   = wceq 953   e. wcel 955  A.wral 1637  E.wrex 1638  Vcvv 1802   (_ wss 2037  (/)c0 2270  P~cpw 2391  U_ciun 2556  Ord word 2937  Oncon0 2938  Lim wlim 2939  suc csuc 2940  ` cfv 3172  R1cr1 4613

This theorem is referenced by:  r1val3 4651

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-r1 4615

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