Theorem r1val1 8202

Description: The value of the cumulative hierarchy of sets function expressed recursively. Theorem 7Q of [Enderton] p. 202. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

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

Distinct variable group:   x, A

Proof of Theorem r1val1

Step	Hyp	Ref	Expression
1		simpr 461	. . . . . 6 |- ( ( A e. dom R1 /\ A = (/) ) -> A = (/) )
2	1	fveq2d 5856	. . . . 5 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = ( R1 ` (/) ) )
3		r10 8184	. . . . 5 |- ( R1 ` (/) ) = (/)
4	2, 3	syl6eq 2498	. . . 4 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = (/) )
5		0ss 3796	. . . . 5 |- (/) C_ U_ x e. A ~P ( R1 ` x )
6	5	a1i 11	. . . 4 |- ( ( A e. dom R1 /\ A = (/) ) -> (/) C_ U_ x e. A ~P ( R1 ` x ) )
7	4, 6	eqsstrd 3520	. . 3 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
8		nfv 1692	. . . . 5 |- F/ x A e. dom R1
9		nfcv 2603	. . . . . 6 |- F/_ x ( R1 ` A )
10		nfiu1 4341	. . . . . 6 |- F/_ x U_ x e. A ~P ( R1 ` x )
11	9, 10	nfss 3479	. . . . 5 |- F/ x ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x )
12		simpr 461	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> A = suc x )
13	12	fveq2d 5856	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ( R1 ` suc x ) )
14		eleq1 2513	. . . . . . . . . . . 12 |- ( A = suc x -> ( A e. dom R1 <-> suc x e. dom R1 ) )
15	14	biimpac 486	. . . . . . . . . . 11 |- ( ( A e. dom R1 /\ A = suc x ) -> suc x e. dom R1 )
16		r1funlim 8182	. . . . . . . . . . . . 13 |- ( Fun R1 /\ Lim dom R1 )
17	16	simpri 462	. . . . . . . . . . . 12 |- Lim dom R1
18		limsuc 6665	. . . . . . . . . . . 12 |- ( Lim dom R1 -> ( x e. dom R1 <-> suc x e. dom R1 ) )
19	17, 18	ax-mp 5	. . . . . . . . . . 11 |- ( x e. dom R1 <-> suc x e. dom R1 )
20	15, 19	sylibr 212	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. dom R1 )
21		r1sucg 8185	. . . . . . . . . 10 |- ( x e. dom R1 -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
22	20, 21	syl 16	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
23	13, 22	eqtrd 2482	. . . . . . . 8 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ~P ( R1 ` x ) )
24		vex 3096	. . . . . . . . . . 11 |- x e. _V
25	24	sucid 4943	. . . . . . . . . 10 |- x e. suc x
26	25, 12	syl5eleqr 2536	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. A )
27		ssiun2 4354	. . . . . . . . 9 |- ( x e. A -> ~P ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
28	26, 27	syl 16	. . . . . . . 8 |- ( ( A e. dom R1 /\ A = suc x ) -> ~P ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
29	23, 28	eqsstrd 3520	. . . . . . 7 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
30	29	ex 434	. . . . . 6 |- ( A e. dom R1 -> ( A = suc x -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) ) )
31	30	a1d 25	. . . . 5 |- ( A e. dom R1 -> ( x e. On -> ( A = suc x -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) ) ) )
32	8, 11, 31	rexlimd 2925	. . . 4 |- ( A e. dom R1 -> ( E. x e. On A = suc x -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) ) )
33	32	imp 429	. . 3 |- ( ( A e. dom R1 /\ E. x e. On A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
34		r1limg 8187	. . . . 5 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) = U_ x e. A ( R1 ` x ) )
35		r1tr 8192	. . . . . . . . 9 |- Tr ( R1 ` x )
36		dftr4 4531	. . . . . . . . 9 |- ( Tr ( R1 ` x ) <-> ( R1 ` x ) C_ ~P ( R1 ` x ) )
37	35, 36	mpbi 208	. . . . . . . 8 |- ( R1 ` x ) C_ ~P ( R1 ` x )
38	37	a1i 11	. . . . . . 7 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` x ) C_ ~P ( R1 ` x ) )
39	38	ralrimivw 2856	. . . . . 6 |- ( ( A e. dom R1 /\ Lim A ) -> A. x e. A ( R1 ` x ) C_ ~P ( R1 ` x ) )
40		ss2iun 4327	. . . . . 6 |- ( A. x e. A ( R1 ` x ) C_ ~P ( R1 ` x ) -> U_ x e. A ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
41	39, 40	syl 16	. . . . 5 |- ( ( A e. dom R1 /\ Lim A ) -> U_ x e. A ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
42	34, 41	eqsstrd 3520	. . . 4 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
43	42	adantrl 715	. . 3 |- ( ( A e. dom R1 /\ ( A e. _V /\ Lim A ) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
44		limord 4923	. . . . . . 7 |- ( Lim dom R1 -> Ord dom R1 )
45	17, 44	ax-mp 5	. . . . . 6 |- Ord dom R1
46		ordsson 6606	. . . . . 6 |- ( Ord dom R1 -> dom R1 C_ On )
47	45, 46	ax-mp 5	. . . . 5 |- dom R1 C_ On
48	47	sseli 3482	. . . 4 |- ( A e. dom R1 -> A e. On )
49		onzsl 6662	. . . 4 |- ( A e. On <-> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
50	48, 49	sylib 196	. . 3 |- ( A e. dom R1 -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
51	7, 33, 43, 50	mpjao3dan 1294	. 2 |- ( A e. dom R1 -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
52		ordtr1 4907	. . . . . . . 8 |- ( Ord dom R1 -> ( ( x e. A /\ A e. dom R1 ) -> x e. dom R1 ) )
53	45, 52	ax-mp 5	. . . . . . 7 |- ( ( x e. A /\ A e. dom R1 ) -> x e. dom R1 )
54	53	ancoms 453	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> x e. dom R1 )
55	54, 21	syl 16	. . . . 5 |- ( ( A e. dom R1 /\ x e. A ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
56		simpr 461	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> x e. A )
57		ordelord 4886	. . . . . . . . . 10 |- ( ( Ord dom R1 /\ A e. dom R1 ) -> Ord A )
58	45, 57	mpan 670	. . . . . . . . 9 |- ( A e. dom R1 -> Ord A )
59	58	adantr 465	. . . . . . . 8 |- ( ( A e. dom R1 /\ x e. A ) -> Ord A )
60		ordelsuc 6636	. . . . . . . 8 |- ( ( x e. A /\ Ord A ) -> ( x e. A <-> suc x C_ A ) )
61	56, 59, 60	syl2anc 661	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> ( x e. A <-> suc x C_ A ) )
62	56, 61	mpbid 210	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> suc x C_ A )
63	54, 19	sylib 196	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> suc x e. dom R1 )
64		simpl 457	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> A e. dom R1 )
65		r1ord3g 8195	. . . . . . 7 |- ( ( suc x e. dom R1 /\ A e. dom R1 ) -> ( suc x C_ A -> ( R1 ` suc x ) C_ ( R1 ` A ) ) )
66	63, 64, 65	syl2anc 661	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> ( suc x C_ A -> ( R1 ` suc x ) C_ ( R1 ` A ) ) )
67	62, 66	mpd 15	. . . . 5 |- ( ( A e. dom R1 /\ x e. A ) -> ( R1 ` suc x ) C_ ( R1 ` A ) )
68	55, 67	eqsstr3d 3521	. . . 4 |- ( ( A e. dom R1 /\ x e. A ) -> ~P ( R1 ` x ) C_ ( R1 ` A ) )
69	68	ralrimiva 2855	. . 3 |- ( A e. dom R1 -> A. x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
70		iunss 4352	. . 3 |- ( U_ x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) <-> A. x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
71	69, 70	sylibr 212	. 2 |- ( A e. dom R1 -> U_ x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
72	51, 71	eqssd 3503	1 |- ( A e. dom R1 -> ( R1 ` A ) = U_ x e. A ~P ( R1 ` x ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   \/ w3o 971   = wceq 1381   e. wcel 1802   A.wral 2791   E.wrex 2792   _Vcvv 3093   C_ wss 3458   (/)c0 3767   ~Pcpw 3993   U_ciun 4311   Tr wtr 4526   Ord word 4863   Oncon0 4864   Lim wlim 4865   suc csuc 4866   dom cdm 4985   Fun wfun 5568   ` cfv 5574   R1cr1 8178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-om 6682  df-recs 7040  df-rdg 7074  df-r1 8180

This theorem is referenced by:  rankr1ai  8214  r1val3  8254