Theorem r1val1 8265

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 462	. . . . . 6 |- ( ( A e. dom R1 /\ A = (/) ) -> A = (/) )
2	1	fveq2d 5885	. . . . 5 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = ( R1 ` (/) ) )
3		r10 8247	. . . . 5 |- ( R1 ` (/) ) = (/)
4	2, 3	syl6eq 2479	. . . 4 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) = (/) )
5		0ss 3793	. . . . 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 3498	. . 3 |- ( ( A e. dom R1 /\ A = (/) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
8		nfv 1755	. . . . 5 |- F/ x A e. dom R1
9		nfcv 2580	. . . . . 6 |- F/_ x ( R1 ` A )
10		nfiu1 4329	. . . . . 6 |- F/_ x U_ x e. A ~P ( R1 ` x )
11	9, 10	nfss 3457	. . . . 5 |- F/ x ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x )
12		simpr 462	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> A = suc x )
13	12	fveq2d 5885	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ( R1 ` suc x ) )
14		eleq1 2495	. . . . . . . . . . . 12 |- ( A = suc x -> ( A e. dom R1 <-> suc x e. dom R1 ) )
15	14	biimpac 488	. . . . . . . . . . 11 |- ( ( A e. dom R1 /\ A = suc x ) -> suc x e. dom R1 )
16		r1funlim 8245	. . . . . . . . . . . . 13 |- ( Fun R1 /\ Lim dom R1 )
17	16	simpri 463	. . . . . . . . . . . 12 |- Lim dom R1
18		limsuc 6690	. . . . . . . . . . . 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 215	. . . . . . . . . 10 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. dom R1 )
21		r1sucg 8248	. . . . . . . . . 10 |- ( x e. dom R1 -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
22	20, 21	syl 17	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
23	13, 22	eqtrd 2463	. . . . . . . 8 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) = ~P ( R1 ` x ) )
24		vex 3083	. . . . . . . . . . 11 |- x e. _V
25	24	sucid 5521	. . . . . . . . . 10 |- x e. suc x
26	25, 12	syl5eleqr 2514	. . . . . . . . 9 |- ( ( A e. dom R1 /\ A = suc x ) -> x e. A )
27		ssiun2 4342	. . . . . . . . 9 |- ( x e. A -> ~P ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
28	26, 27	syl 17	. . . . . . . 8 |- ( ( A e. dom R1 /\ A = suc x ) -> ~P ( R1 ` x ) C_ U_ x e. A ~P ( R1 ` x ) )
29	23, 28	eqsstrd 3498	. . . . . . 7 |- ( ( A e. dom R1 /\ A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
30	29	ex 435	. . . . . 6 |- ( A e. dom R1 -> ( A = suc x -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) ) )
31	30	a1d 26	. . . . 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 2906	. . . 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 430	. . 3 |- ( ( A e. dom R1 /\ E. x e. On A = suc x ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
34		r1limg 8250	. . . . 5 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) = U_ x e. A ( R1 ` x ) )
35		r1tr 8255	. . . . . . . . 9 |- Tr ( R1 ` x )
36		dftr4 4523	. . . . . . . . 9 |- ( Tr ( R1 ` x ) <-> ( R1 ` x ) C_ ~P ( R1 ` x ) )
37	35, 36	mpbi 211	. . . . . . . 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 2837	. . . . . 6 |- ( ( A e. dom R1 /\ Lim A ) -> A. x e. A ( R1 ` x ) C_ ~P ( R1 ` x ) )
40		ss2iun 4315	. . . . . 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 17	. . . . 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 3498	. . . 4 |- ( ( A e. dom R1 /\ Lim A ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
43	42	adantrl 720	. . 3 |- ( ( A e. dom R1 /\ ( A e. _V /\ Lim A ) ) -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
44		limord 5501	. . . . . . 7 |- ( Lim dom R1 -> Ord dom R1 )
45	17, 44	ax-mp 5	. . . . . 6 |- Ord dom R1
46		ordsson 6630	. . . . . 6 |- ( Ord dom R1 -> dom R1 C_ On )
47	45, 46	ax-mp 5	. . . . 5 |- dom R1 C_ On
48	47	sseli 3460	. . . 4 |- ( A e. dom R1 -> A e. On )
49		onzsl 6687	. . . 4 |- ( A e. On <-> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
50	48, 49	sylib 199	. . 3 |- ( A e. dom R1 -> ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) )
51	7, 33, 43, 50	mpjao3dan 1331	. 2 |- ( A e. dom R1 -> ( R1 ` A ) C_ U_ x e. A ~P ( R1 ` x ) )
52		ordtr1 5485	. . . . . . . 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 454	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> x e. dom R1 )
55	54, 21	syl 17	. . . . 5 |- ( ( A e. dom R1 /\ x e. A ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
56		simpr 462	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> x e. A )
57		ordelord 5464	. . . . . . . . . 10 |- ( ( Ord dom R1 /\ A e. dom R1 ) -> Ord A )
58	45, 57	mpan 674	. . . . . . . . 9 |- ( A e. dom R1 -> Ord A )
59	58	adantr 466	. . . . . . . 8 |- ( ( A e. dom R1 /\ x e. A ) -> Ord A )
60		ordelsuc 6661	. . . . . . . 8 |- ( ( x e. A /\ Ord A ) -> ( x e. A <-> suc x C_ A ) )
61	56, 59, 60	syl2anc 665	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> ( x e. A <-> suc x C_ A ) )
62	56, 61	mpbid 213	. . . . . 6 |- ( ( A e. dom R1 /\ x e. A ) -> suc x C_ A )
63	54, 19	sylib 199	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> suc x e. dom R1 )
64		simpl 458	. . . . . . 7 |- ( ( A e. dom R1 /\ x e. A ) -> A e. dom R1 )
65		r1ord3g 8258	. . . . . . 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 665	. . . . . 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 3499	. . . 4 |- ( ( A e. dom R1 /\ x e. A ) -> ~P ( R1 ` x ) C_ ( R1 ` A ) )
69	68	ralrimiva 2836	. . 3 |- ( A e. dom R1 -> A. x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
70		iunss 4340	. . 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 215	. 2 |- ( A e. dom R1 -> U_ x e. A ~P ( R1 ` x ) C_ ( R1 ` A ) )
72	51, 71	eqssd 3481	1 |- ( A e. dom R1 -> ( R1 ` A ) = U_ x e. A ~P ( R1 ` x ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   \/ w3o 981   = wceq 1437   e. wcel 1872   A.wral 2771   E.wrex 2772   _Vcvv 3080   C_ wss 3436   (/)c0 3761   ~Pcpw 3981   U_ciun 4299   Tr wtr 4518   dom cdm 4853   Ord word 5441   Oncon0 5442   Lim wlim 5443   suc csuc 5444   Fun wfun 5595   ` cfv 5601   R1cr1 8241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-r1 8243

This theorem is referenced by:  rankr1ai  8277  r1val3  8317